Clifford algebra as an adjunction? Background
For definiteness (even though this is a categorical question!) let's agree that a vector space is a finite-dimensional real vector space and that an associative algebra is a finite-dimensional real unital associative algebra.
Let $V$ be a vector space with a nondegenerate symmetric bilinear form $B$ and let $Q(x) = B(x,x)$ be the associated quadratic form.  Let's call the pair $(V,Q)$ a quadratic vector space.
Let $A$ be an associative algebra and let's say that a linear map $\phi:V \to A$ is Clifford if
$$\phi(x)^2 = - Q(x) 1_A,$$
where $1_A$ is the unit in $A$.
One way to define the Clifford algebra associated to $(V,Q)$ is to say that it is universal for Clifford maps from $(V,Q)$.  Categorically, one defines a category whose objects are pairs $(\phi,A)$ consisting of an associative algebra $A$ and a Clifford map $\phi: V \to A$ and whose arrows
$$h:(\phi,A)\to (\phi',A')$$
are morphisms $h: A \to A'$ of associative algebras such that the obvious triangle commutes:
$$h \circ \phi = \phi'.$$
Then the Clifford algebra of $(V,Q)$ is the universal initial object in this category.  In other words, it is a pair $(i,Cl(V,Q))$ where $Cl(V,Q)$ is an associative algebra and $i:V \to Cl(V,Q)$ is a Clifford map, such that for every Clifford map $\phi:V \to A$, there is a unique morphism
$$\Phi: Cl(V,Q) \to A$$
extending $\phi$; that is, such that $\Phi \circ i = \phi$.
(This is the usual definition one can find, say, in the nLab.)
Question
I would like to view the construction of the Clifford algebra as a functor from the category of quadratic vector spaces to the category of associative algebras.  The universal property says that if $(V,Q)$ is a quadratic vector space and $A$ is an associative algebra, then there is a bijection of hom-sets
$$\mathrm{hom}_{\mathbf{Assoc}}(Cl(V,Q), A) \cong \mathrm{cl-hom}(V,A)$$
where the left-hand side are the associative algebra morphisms and the right-hand side are the Clifford morphisms.
My question is whether I can view $Cl$ as an adjoint functor in some way.  In other words, is there some category $\mathbf{C}$ such that the right-side is
$$\mathrm{hom}_{\mathbf{C}}((V,Q), F(A))$$
for some functor $F$ from associative algebras to $\mathbf{C}$.  Naively I'd say $\mathbf{C}$ ought to be the category of quadratic vector spaces, but I cannot think of a suitable $F$.
I apologise if this question is a little vague.  I'm not a very categorical person, but I'm preparing some notes for a graduate course on spin geometry next semester and the question arose in my mind.
 A: This answer builds on sdcvvc's answer and the comments below it, and in particular concerns the (non)existence of a canonical quadratic form $q$ (in sdcvvc's notation).
Let me denote by $\mathcal{Q}$ the category of quadratic real vector spaces (where the symmetric bilinear form is not necessarily nondegenerate), and by $\mathcal{A}$ some subcategory of the category $\mathcal{A}ss$ of finite-dimensional real unital associative algebras that contains the image of the Clifford functor $\mathcal{C}l: \mathcal{Q} \to \mathcal{A}ss$.
Notice that $\mathcal{Q}$ contains $\mathrm{\mathbf{Vect}}_\mathbb{R}$ as the full subcategory whose objects of the form $(V, 0)$, and that the restriction of the functor $\mathcal{C}l: \mathcal{Q} \to \mathcal{A}$ to this subcategory is the exterior algebra functor $V \mapsto \Lambda^{\ast}V$. Then,
$$\mathrm{Hom}_{\mathcal{A}}(\Lambda^\ast V, A) \cong \lbrace \phi: V \to A \; | \; \phi(v)^2 = 0 \rbrace$$
You can make $\Lambda^{\ast}(-)$ into a left adjoint by restricting $\mathcal{A}$ to be the category of $\mathbb{Z}_2$-graded supercommutative algebras (maybe you can take a bigger subcategory?). The right adjoint should then be the functor taking such an algebra to its odd-degree part considered as a vector space. This makes the Clifford condition $\phi(v)^2 = 0$ trivially true.
It is the latter observation the one that allows us to cook up such an $\mathcal{A}$. However, in the general case the Clifford condition does involve the quadratic form on the vector space that is the domain, and so it doesn't seem possible to me that we could do something like the above universally.
A: If I understand the definitions correctly:
Let $C$ be the category of pairs (V,q) where V is a vector space on a fixed field and q is a quadratic form. A morphism $f: (V,q) \rightarrow (V',q')$ is a linear map $V \to V'$ preserving the quadratic form.
Let $D$ be the category of unital algebras over the field. Morphisms are linear maps preserving multiplication and identity.
We've got a forgetful functor $D \rightarrow C$ that maps an algebra V to the quadratic vector space $ (V,q)$ where $q(x)=(x \cdot x) \cdot 1$. This functor has as left adjoint the Clifford algebra construction.
(I'm inexperienced, so this might be plain wrong. But surely an adjoint functor is hiding here.)
A: UPDATE: the following argument is wrong, see the comments.
If $\mathcal{C}l$ admits a right adjoint then it preserves colimits, and coproducts in particular. Now, in your category of quadratic vector spaces, the coproduct of $(V, Q)$ and $(V', Q')$ is $(V \oplus V', Q \oplus Q')$; for associative algebras $A$ and $A'$, its coproduct is given by tensor product over $\mathbb{R}$. Hence, it is necessary that $$\mathcal{C}l(V \oplus V', Q \oplus Q') \cong \mathcal{C}l(V, Q) \otimes_{\mathbb{R}} \mathcal{C}l(V', Q')$$
Here's a counterexample: take $V = V' = \mathbb{R}$ with $Q = Q' = -1$. By the classification of Clifford algebras, we know that $\mathcal{C}l(\mathbb{R}, -1) \cong \mathbb{C}$ and $\mathcal{C}l(\mathbb{R}^2, \mathrm{diag}(-1,-1)) \cong \mathbb{H}$. It is now enough to observe that $$\mathbb{H} \not\cong \mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} $$
A: I'm hoping for a second opinion on this question. The same question occurred to me, and google led me to this thread. At first glance, the consensus answer here (there is no right-adjoint to $Cl$) seems a plausibly argued. But after some thought, I'm not convinced.
We know that a universal construction, if it exists for every object in the source category, always gives an adjunction between categories.
An object satisfying the universal property for a Clifford algebra can be explicitly constructed from any vector space with quadratic form as a quotient of the tensor algebra. So an object satisfying the universal property always exists, therefore it is a left-adjoint. And what should the right-adjoint functor to the Clifford functor? Why nothing other than the underlying map from associative algebras to quadratic spaces, with quadratic form $q(x)=x^2$. This is the only possible quadratic form on the underlying vector spaces which will make the stipulation in the universal construction about the linear maps into morphisms in the category of quadratic vector spaces.
I should conclude that the right-adjoint of $Cl$ is a forgetful functor $k\text{-Alg}\to k\text{-Quad}$ which takes an associative algebra and forgets multiplication but remembers how to square vectors. The unit of this adjunction is the Clifford algebra structure map, and the counit is the map from the Clifford algebra on the quadratic vector space underlying any algebra $A$ to $A$ which takes $a_1\cdot a_2\mapsto a_1a_2$.
This is of course exactly the unaccepted answer that sdcvvc gives above, though without much detail. Qiaochu Yuan says that the claimed quadratic form $q(x)=x^2$ on the underlying vector space of an associative algebra is not actually quadratic. I cannot see why not. Why is sdcvvc's answer incorrect?
Alberto García-Raboso gives an answer as well, where in the discussion it is settled that $Cl$ preserves finite coproducts. If we can also show that it preserves cokernels then we know that it must have a right-adjoint, by Freyd adjoint functor theorem, right?
And have I misunderstood the relationships between universal morphisms and adjunctions? Is it not the case that we can simply read off the adjoint functor out of the universal property?
And do we really need to consider, as Andrew Stacy suggests, some kind of pointed vector spaces? If so, why?
I wanted to post my questions as comments, not an answer, but I guess I don't have enough rep. Please forgive me.
A: Disqualifier: this isn't a complete answer.
There's a basic "chalk and cheese" problem here.  The "categories" that you are comparing are of two different types, although they do seem similar on the surface.  On the one hand you have an honest algebraic category: that of associative algebras.  But the other category (which, admittedly, is not precisely defined) is "vector spaces plus quadratic forms".  This is not algebraic (over Set).  There's no "free vector space with a non-degenerate quadratic form" and there'll (probably) be lots of other things that don't quite work in the way one would expect for algebraic categories.  For example, as you require non-degeneracy, all morphisms have to be injective linear maps which severely limits them.  You could add degenerate quadratic forms (which means, as AGR hints, that you regard exterior algebras as a sort of degenerate Clifford algebra - not a bad idea, though!) but this still doesn't get algebraicity: the problem is that the quadratic form goes out of the vector space, not into it, so isn't an "operation".
However, you may get some mileage if you work with pointed objects.  I'm not sure of my terminology here, but I mean that we have a category $\mathcal{C}$ and some distinguished object $C_0$ and consider the category $(C,\eta,\epsilon)$ where $\eta : C_0 \to C$, $\epsilon : C \to C_0$ are such that $\epsilon \eta = I_{C_0}$.  In Set, we take $C_0$ as a one-point set.  In an algebraic category, we take $C_0$ as the free thing on one object.  Then the corresponding pointed algebraic category is algebraic over the category of pointed sets (I think!).
The point (ha ha) of this is that in the category of pointed associative algebras one does have a "trace" map: $\operatorname{tr} : A \to \mathbb{R}$ given by $(a,b) \mapsto \epsilon(a \cdot b)$.  Thus one should work in the category of pointed associative $\mathbb{Z}/2$-graded algebras whose trace map is graded symmetric.
In the category of pointed vector spaces, one can similarly define quadratic forms as operations.  You need a binary operation $b : |V| \times |V| \to |V|$ (only these products are of pointed sets) and the identity $\eta \epsilon b = b$ to ensure that $b$ really lands up in the $\mathbb{R}$-component of $V$ (plus symmetry).
Whilst adding the pointed condition is non-trivial for algebras, it is effectively trivial for vector spaces since there's an obvious functor from vector spaces to pointed vector spaces, $V \mapsto V \oplus \mathbb{R}$ that is an equivalence of categories.
Assuming that all the $\imath$s can be crossed and all the $l$s dotted, the functor that you want is now the forgetful functor from pointed associative algebras to pointed quadratic vector spaces.
A: Here is another proposal. The key will be to enlarge the category of quadratic vector spaces fairly substantially. Here are three hints leading towards the proposal:


*

*Clifford algebras can and should be thought of as $\mathbb{Z}_2$-graded; for example, Clifford algebras over $\mathbb{R}$ give every element of the $\mathbb{Z}_2$-graded Brauer group / Brauer-Wall group of $\mathbb{R}$.

*Thinking of Clifford algebras as deformations of exterior algebras, the analogous deformations of symmetric algebras are the Weyl algebras. It is possible to combine the construction of exterior algebras and symmetric algebras into a single construction, namely the construction of the symmetric algebra on a super vector space.

*Weyl algebras are almost universal enveloping algebras of certain Lie algebras. More precisely, let $(V, \omega)$ be a symplectic vector space. From this data we can construct a Lie algebra $V \oplus \mathbb{R}$ such that $\mathbb{R}$ is central and $[v, w] = \omega(v, w) \in \mathbb{R}$ where $v, w \in V$. Then the Weyl algebra constructed from $(V, \omega)$ is the quotient of the universal enveloping algebra $U(V \oplus \mathbb{R})$ by the extra relation that $1 \in \mathbb{R}$ acts as the identity. 


So let's try this: 

There is a forgetful functor from super algebras to a certain category of super Lie algebras with extra structure whose left adjoint restricts to 1) the symmetric algebra functor, 2) the exterior algebra functor, 3) the Weyl algebra functor, 4) the Clifford algebra functor, and 5) the universal enveloping algebra functor on suitable subcategories. 

This left adjoint generalizes every functor discussed above. Some details: 
The first category $\text{SAlg}$, the category of super algebras, is the category of monoid objects in super vector spaces. As a category it can be thought of as the category of $\mathbb{Z}_2$-graded algebras, but as a symmetric monoidal category it has a nontrivial braiding given by the Koszul sign rule as usual. In particular, a commutative super algebra is commutative in the super sense, not the usual sense.
The second category begins from the category $\text{SLieAlg}$ of super Lie algebras, which is the category of Lie algebra objects in super vector spaces; here it's quite important to distinguish this category from the category of $\mathbb{Z}_2$-graded Lie algebras because the Koszul sign rule changes some signs in the Lie algebra axioms. In particular, the skew-symmetry axiom becomes
$$[x, y] = - (-1)^{|x| |y|} [y, x]$$
for homogeneous elements $x, y$. Hence if either $x$ or $y$ is even then this is skew-symmetry in the usual sense, but if $x$ and $y$ are both odd then we actually have symmetry. This is crucial.
The category we're actually interested is not quite this category; instead it is the category of super Lie algebras $\mathfrak{g}$ equipped with a morphism $\mathbb{R}[0] \to \mathfrak{g}$, where $\mathbb{R}[0]$ denotes the abelian Lie algebra $\mathbb{R}$ in degree $0$, with central image. This might be called the category of "centrally pointed super Lie algebras," maybe. 
The forgetful functor from $\text{SAlg}$ to the above category sends a super algebra $A$ to the underlying super vector space of $A$ equipped with the super Lie bracket
$$[x, y] = xy - (-1)^{|x| |y|} yx$$
on homogeneous elements, with the map $\mathbb{R}[0] \to A$ being given by the unit element of $A$.
The left adjoint to this forgetful functor sends $\mathbb{R}[0] \to \mathfrak{g}$ to the quotient of the universal enveloping (super) algebra $U(\mathfrak{g})$ by the extra relation that $1 \in \mathbb{R}[0]$ acts as the identity. I'll also call this functor $U$. Here are the five promised subcategories (they are not full subcategories):


*

*Given a vector space $V$, send it to the abelian super Lie algebra $V[0] \oplus \mathbb{R}[0]$. Then $U(V[0] \oplus \mathbb{R}[0])$ is the symmetric algebra on $V$. 

*Given a vector space $V$, send it to $V[1] \oplus \mathbb{R}[0]$. Then $U(V[1] \oplus \mathbb{R}[0])$ is the exterior algebra on $V$.

*Given a symplectic vector space $(V, \omega)$, send it to $V[0] \oplus \mathbb{R}[0]$ with bracket given by $\omega$ as previously discussed. Then $U(V[0] \oplus \mathbb{R}[0])$ is the Weyl algebra of $(V, \omega)$.

*Given a quadratic vector space $(V, Q)$, let $B(v, w) = Q(v + w) - Q(v) - Q(w)$ be twice the symmetric bilinear form determined by $Q$ and send it to $V[1] \oplus \mathbb{R}[0]$ with bracket given by $B$, as previously discussed for the Weyl algebra. Then $U(V[1] \oplus \mathbb{R}[0])$ is the Clifford algebra of $(V, Q)$.

*Given a Lie algebra $\mathfrak{g}$, send it to $\mathfrak{g}[0] \oplus \mathbb{R}[0]$. Then $U(\mathfrak{g}[0] \oplus \mathbb{R}[0])$ is the usual universal enveloping algebra of $\mathfrak{g}$.

