"Quasi-coherent" vector spaces in Sch/S $\DeclareMathOperator\Vec{Vec}\newcommand\Sch{\mathrm{Sch}}\DeclareMathOperator\Hom{Hom}$Let $S$ be a base scheme. Let me write $\Vec(S)$ to denote the category of $\mathbb A_S$-vector space objects internal to the category $\Sch_{/S}$ of $S$-schemes. For each such vector space $p:V\to S$ (or bundle from the perspective of $\Sch$) we get a sheaf$$U\mapsto \Hom_{\Vec(U)}(p\rvert_U,\mathbb A_U)$$
on $S$ which I denote by $L_S(p)$. Here $p\rvert_U$ is the $\mathbb A_U$-vector space in $\Sch/U$ obtained by restricting $p$ and the structure maps of $p$ accordingly. The sheaf $L_S(p)$ is naturally a sheaf of $\mathcal O_S$-modules.
Question: Is $L_S(p)$ always quasicoherent? Is there a nice characterization of those internal $\mathbb A_S$-vector spaces $p$ for which the associated sheaf $L_S(p)$ of linear functions is quasicoherent?
I have a partial result, but I have no idea how I can attack the general case. If the underlying $S$-bundle map $p$ is affine, then the sheaf $L_S(p)$ will be quasicoherent. The argument (with the help of synthetic differential geometry) goes like this. The $\mathbb A_S$-vector space $\mathbb A_S$ is euclidean. This has a technical meaning in SDG: basically $\mathbb A_S$ is microlinear and satisfies a generalized version of the KL-axiom. It then follows that a map $V\to \mathbb A_S$ of $S$-schemes out of any $\mathbb A_S$-vector space $V$ is already $\mathbb A_S$-linear if it just preserves the $\mathbb A_S$-scalar action (see Lavendhomme first chapter). This means that the sheaf $L_S(p)$ has a simpler description (even when $p$ is not affine). The set $L_S(p)(U)$ consists of those $U$-scheme maps $p\rvert_U\to \mathbb A_U$ which preserve the $\mathbb A_U$-action. A $(\mathbb A_S,\cdot)$-monoid action on an affine $S$-scheme $p:V\to S$ is the same thing as an $\mathbb N$-grading of the $\mathcal O_S$-algebra $p_*\mathcal O_V$, and we find that under this correspondence $L_S(p)$ is precisely the $1$-graded piece of $p_*\mathcal O_S$. Thus, $L_S(p)=(p_*\mathcal O_V)_1$ is quasicoherent.
Edit. Here is an explanation of what an $\mathbb A_S$-vector space is. Let $\mathbb C$ be a category with finite limits and let $R$ be a commutative ring object in $\mathbb C$. This means that $R$ comes equipped with structure maps $0:\mathbf 1 \to R$, $1:\mathbf 1\to R$, $+:R\times R\to R$, etc. which satisfy the axioms of a commutative ring, but formulated in terms of diagrams so that they make sense in $\mathbb C$. An internal $R$-module in $\mathbb C$ is an object $V$ together with morphisms $+:V\times V\to V$, $0:\mathbf 1\to V$ and $\cdot: R\times V\to V$ such that the axioms of a module are satisfied, translated suitably into diagramatic form. An $\mathbb A_S$-vector space is an $\mathbb A_S$-module in the category $\Sch_{/S}$. I call it a vector space, because the ring object $\mathbb A_S$ satisfies$$\Sch_{/S}\models \forall x:\mathbb A_S.\, (\neg(x = 0)\to \exists ! y:\mathbb A_S. x\cdot y= 1)$$
in the internal language of $\Sch_{/S}$, but this has no influence on the content of my question! :)
From the perspective of $\Sch$, a $\mathbb A_S$-vector space looks more like a "bundle with a fiberwise vector space structure". It is an $S$-scheme $p:V\to S$ together with morphisms $\mathbb A\times V\to V$, $+:V\times_SV\to V$ and $0:S\to V$ above $S$ such that the correct diagrams commute.
 A: What I wrote in the first comment above is wrong.  I usually work with "projective Abelian cones" rather than "Abelian cones", and projective Abelian cones (typically) do not have a section.  That makes a huge difference.
The sheaf defined by the OP agrees with the pullback by the zero section of the sheaf of relative differentials of the morphism $p$ from the Abelian cone $V$ to the base scheme $S$.  The sheaf of relative differentials is always quasi-coherent.  Pullback preserves quasi-coherent sheaves.  So the pullback by zero section of the sheaf of relative differentials of $p$ is quasi-coherent.
Edit.  For an Abelian cone over $S$, $p:V\to S$,
the scheme $V$ is a group scheme over $S$ under addition, and the geometric fibers are connected.  Denote by $\Omega_{V/S}$ the sheaf of relative differentials.  Denote by $\Omega_{V,S}$ the pullback of $\Omega_{V/S}$ by the zero section.  By the same basic argument that proves, for each group scheme, the equivalence of the tangent space at the identity with the vector space of left-invariant tangent vector fields, there is a natural isomorphism, $$i_{V/S}: \Omega_{V/S}\xrightarrow{\cong} p^*\Omega_{V,S}.$$  In particular, $\Omega_{\mathbb{A}^1_S,S}$ is a free $\mathcal{O}_S$-module of rank $1$, and the restriction of $i_{\mathbb{A}^1_S/S}$ to the open subscheme $\mathbb{G}_{m,S}\subset \mathbb{A}^1_S$ shows that $\Omega_{\mathbb{A}^1_S,S}$ also pulls back to the sheaf of relative differentials $\Omega_{\mathbb{G}_{m,S}/S}$.
The action of $\mathbb{G}_m$ on $V$ over $S$ induces a natural $\mathcal{O}_V$-module homomorphism, $$j_{V/S}:\Omega_{V/S}\to p^*\Omega_{\mathbb{A}^1_S,S}.$$
Note that the action of $\mathbb{G}_m$ on $V$ naturally "linearizes" to an action of $\mathbb{G}_{m,S}$ on $\Omega_{V/S}$. The zero section is fixed by the action of $\mathbb{G}_m$.  Thus, there is an induced linearization of $\mathbb{G}_m$ on $\Omega_{V,S}$.  Of course every $\mathcal{O}_S$-module has a tautological linearization, where $\mathbb{G}_m$ acts by (usual) scaling.  In characteristic $0$, the two actions automatically agree, but this can fail in positive characteristic (this is what my comment above is about).
Hypothesis. Assume that the induced $\mathbb{G}_m$-linearization agrees with the tautological linearization on $\Omega_{V,S}$.
In this case, the map $j_{V/S}$ together with the isomorphism $i_{V/S}$ give a homomorphism of $\mathcal{O}_V$-modules, $$j_{V/S}\circ i_{V/S}^{-1}:p^*\Omega_{V,S}\to p^*\Omega_{\mathbb{A}^1_S,S}.$$  There is an Abelian cone over $S$, $q:W\to S$, and a universal $\mathcal{O}_W$-module homomorphism, $$k_{W/S}: q^*\Omega_{V,S}\to q^*\Omega_{\mathbb{A}^1_S,S},$$ namely, $$W=\text{Spec}_S \left(\text{Sym}^\bullet_{\mathcal{O}_S} \ \Omega'_{V,S} \right), \ \ \Omega'_{V,S}:=\Omega_{\mathbb{A}^1_S,S}^\vee\otimes_{\mathcal{O}_S}\Omega_{V,S}.$$  Thus, there is an induced $S$-morphism, $$f:V\to W.$$  It is straightforward to use the functoriality of $\Omega_{V/S}$ with respect to addition and scaling to check that $f$ is a morphism of Abelian cones over $S$.
By construction, the sheaf of relative differentials of $f$ is zero, i.e.,  the morphism $f$ is formally unramified.  I believe that the morphism $f$ is always an isomorphism, but this is not necessary to answer the question by the OP.  The sections of $\Omega'_{V,S}$ are equivalent to the homomorphisms of $\mathbb{Z}_{\geq 0}$-graded $\mathcal{O}_S$-algebras, $$\text{Sym}^\bullet_{\mathcal{O}_S} \mathcal{O}_S \to \text{Sym}^\bullet_{\mathcal{O}_S} \Omega'_{V,S}.$$  Applying relative Spec, this is equivalent to a morphism of Abelian cones over $S$ (the hypothesis above is needed to prove that the $S$-morphism respects scaling by $\mathbb{G}_m$), $$W\to \mathbb{A}^1_S.$$  By precomposing with $f$, the sections of $\Omega_{V,S}$ give morphisms of Abelian cones over $S$, $$V\to W \to \mathbb{A}^1_S.$$
Of course, for every morphism of Abelian cones over $S$, $$g:V\to \mathbb{A}^1_S,$$ functoriality of $\Omega$ gives an induced homomorphism of $\mathcal{O}_V$-modules, $$p^*\Omega_{\mathbb{A}^1_S,S}\to p^*\Omega_{V,S}.$$  Pulling back by the zero section gives a section of $\Omega'_{V,S}$.
A: This is not a complete answer, but it explains the affine case in a bit more elementary terms. (Also because I do not understand what is going on in Jason's answer.)
Let us first assume that $S = \mathrm{Spec}(R)$ and $V = \mathrm{Spec}(A)$ are affine, so $A$ is a commutative $R$-algebra.
That $V$ carries the structure of a commutative group internal to $\mathbf{Sch}_R$ means that $A$ carries the structure of a cocommutative cogroup internal to $\mathbf{CAlg}_R$. That is, we have bicommutative (i.e., commutative and cocommutative) Hopf algebra $A$ over $R$.
That $V$ carries an action of the monoid $\mathbb{A}^1_R$ internal to $\mathbf{Sch}_R$ means that $A$ carries a coaction from the coalgebra $R[T]$. If $\ell : A \to A[T]$ is the coaction, one shows that $A$ becomes a commutative $\mathbb{N}$-graded $R$-algebra with $A_n = \{a \in A:  \ell(a) = a T^n \}$, and that every grading arises this way.
Now, an $\mathbb{A}^1_R$-module structure on $V$ internal to $\mathbf{Sch}_R$ has both: we have a bicommutative Hopf algebra structure on $A$ as well as an $\mathbb{N}$-grading on $A$. For a module, we have left two compatibility conditions: $r \cdot (x+y) = r \cdot x + r \cdot y$ and $(r + s) \cdot x = r \cdot x + s \cdot x$. The first one translates to the condition that the comultiplication $\Delta : A \to A \otimes_R A$ is graded. The second condition is a bit more complicated to write down (right now, I don't have anything to offer except for ugly equations with Sweedler notation), but we don't need this for the following anyway.
The morphisms of $R$-schemes $V \to \mathbb{A}^1_R$ are just homomorphisms of $R$-algebras $R[T] \to A$, i.e. elements of $A$. A morphism $V \to \mathbb{A}^1_R$ is compatible with the $\mathbb{A}^1_R$-module structure iff it is compatible with the $\mathbb{A}^1$-action and the addition. So these correspond to homomorphisms of $R$-algebras $R[T] \to A$ which are graded and compatible with the comultiplication. And these correspond to elements $a \in A$ which are homogenous of degree $1$ and satisfy $\Delta(a) = a \otimes 1 + 1 \otimes a$. As was already mentioned by the OP, the second condition follows from the first, but actually we will not need this for what follows.
So this describes the $R$-module $L(V)(S)$. More generally, for a basic-open subset $D(f) \subseteq \mathrm{Spec}(R)$ we see that $L(V)(D(f))$ consists of the elements of degree $1$ in $A_f$ with $\Delta(a) = a \otimes 1 + 1 \otimes a$ (with the induced grading and comultiplication on $A_f$). This is a $R_f$-submodule of $A_f$. So in particular the canonical map
$$L(V)(S)_f \to L(V)(D(f))$$
is injective, and what remains is to show surjectivity. There is an elementary argument for this:
Let $a/f^k \in L(V)(D(f)) \subseteq A_f$ be as above. Let $a = \sum_i a_i$ be the homogeneous decomposition in $A$. Choose some $N \geq 0$ such that $f^N a_0 = 0$ and $f^N a_i = 0$ for $i > 1$. Then $f^N a \in A$ is homogeneous of degree $1$ and $(f^N a) / f^{k+N} = a / f^k$, so we can actually assume that $a$ is already homogeneous of degree $1$. Next, since $\Delta(a/f^k) = a/f^k \otimes 1 + 1 \otimes a/f^k$ holds in $(A \otimes A)_f$, there is some $N \geq 0$ such that $f^N (\Delta(a) - a \otimes 1 - 1 \otimes a) = 0$ holds in $A \otimes A$. Then we can use $f^N a$ and we are done.
If $S$ is not necessarily affine, but $p : V \to S$ is affine, then $L(V)$ is a quasi-coherent $\mathcal{O}_S$-module (just glue along the affine parts in $S$).
Here is how I would approach the (almost) general case: Notice that $L(V)$ is always a submodule of $p_* \mathcal{O}_V$, which maps $U \subseteq S$ to the set of all morphisms $V_U \to \mathbb{A}^1_U$. Let us assume that $p$ is quasi-compact and quasi-separated, so that $p_* \mathcal{O}_V$ is quasi-coherent. First consider only the additive structure (and morphisms preserving it). Then consider only the $\mathbb{A}^1_S$-action (and morphisms preserving it). If this both works out, $L(V)$ is the intersection of two quasi-coherent modules, hence quasi-coherent.
If $V$ carries the trivial $\mathbb{A}^1$-action (so it is not a module), notice that the sheaf of $\mathbb{A}^1$-morphisms $V \to \mathbb{A}^1$ is equal to $p_* \mathcal{O}_V$. So we really want to assume that $p$ is quasi-compact and quasi-separated.
A: The claim is true as long as $p$ is qc qs. This is needed so that $p_*$ preserves quasi-coherent modules. Notice that, by using the isomorphisms $\mathrm{Hom}_U(V_U,\mathbb{A}^1_U) \cong \Gamma(V_U,\mathcal{O}_{V_U}) = \Gamma(p^{-1}(U),\mathcal{O}_V)$, the module $L$ is isomorphic to a submodule of $p_* \mathcal{O}_V$.
Curiously, we do not need to assume any vector space axioms - only the structure matters. We may start with any $S$-morphism $\mathbb{A}^1_S \times_S V \to V$ and will prove below that the submodule $L_1$ of $\mathbb{A}^1$-equivariant morphisms $V \to \mathbb{A}^1_S$ (i.e., the module which maps $U \subseteq S$ to the set of $\mathbb{A}^1_U$-equivariant morphisms $V_U \to \mathbb{A}^1_U$) is a quasi-coherent submodule of $p_* \mathcal{O}_V$. Then, if we start with any $S$-morphism $V \times_S V \to V$ ("addition") we will show that the submodule $L_2$ of additive morphisms $V \to \mathbb{A}^1_S$ is also quasi-coherent. So, if $p : V \to S$ carries a vector space structure, $L = L_1 \cap L_2$ is the intersection of two quasi-coherent modules inside of a quasi-coherent module, thus it's also quasi-coherent.
Let $\alpha : \mathbb{A}^1_S \times_S V \to V$ be an $S$-morphism. If $U \subseteq S$ is open, then a morphism $f : V_U \to \mathbb{A}^1_U$ is $\mathbb{A}^1_U$-equivariant iff $m \circ (\mathbb{A}^1_U \times f) = f \circ \alpha$, where $m : \mathbb{A}^1_U \times_U \mathbb{A}^1_U \to \mathbb{A}^1_U$ is the multiplication. The morphism $f$ corresponds to a local section $x \in \Gamma(p^{-1}(U),\mathcal{O}_V)$, and the condition is saying that $\alpha^{\#}(x) = T \boxtimes x$ holds, where $T$ is the canonical global section of $\mathbb{A}^1$ and $\boxtimes$ is the canonical map $\Gamma(X,\mathcal{O}_X) \otimes \Gamma(Y,\mathcal{O}_Y) \to \Gamma(X \times Y,\mathcal{O}_{X \times Y})$ for $X = \mathbb{A}^1_U$, $Y = V_U$.
The morphism $\alpha^{\#} : \mathcal{O}_V \to \alpha_* \mathcal{O}_{\mathbb{A}^1_S \times_S V}$ induces a morphism $u := p_* \alpha^{\#} : p_*  \mathcal{O}_V \to p_*  \alpha_* \mathcal{O}_{\mathbb{A}^1_S \times_S V}$. Since $\alpha$ is an $S$-morphism, we have $ p_*  \alpha_* \mathcal{O}_{\mathbb{A}^1_S \times_S V} = p_* (\mathrm{pr}_2)_*  \mathcal{O}_{\mathbb{A}^1_S \times_S V}$. Since $\mathrm{pr}_2 : \mathbb{A}^1_S \times_S V \to V$ is affine and hence qs qs, we see that the latter module is quasi-coherent.
We define $v' : \mathcal{O}_V  \to (\mathrm{pr}_2)_*  \mathcal{O}_{\mathbb{A}^1_S \times_S V}$ via multiplication with $T$ (if everything is affine, this is just $R \to R[T]$, $r \mapsto rT$), and then $v := p_* v'$.
Now we see that $L_1$ is precisely the equalizer of $u$ and $v$. Hence, $L_1$ is quasi-coherent.
The approach for additivity is basically the same as above. (Maybe there is even a common generalization.)
Let $\mu : V \times_S V \to V$ be any $S$-morphism, so $p \mu = p \mathrm{pr}_2$. We define $u := p_* \mu^{\#} : p_* \mathcal{O}_V \to p_* \mu_* \mathcal{O}_{V \times_S V} = p_* (\mathrm{pr}_2)_* \mathcal{O}_{V \times_S V}$. Then we define $v' := \mathcal{O}_V \to (\mathrm{pr}_2)_* \mathcal{O}_{V \times_S V}$ by $v'(x) = x \boxtimes 1 + 1 \boxtimes x$, and $v := p_* v'$. Then $L_2$ is the equalizer of $u,v$.
