Start with a quadratic form $q$ on a vector space $V$. A module $M$ over the corresponding Clifford algebra is determined by a map $\cdot:V\otimes M\to M$ satisfying $v\cdot(v\cdot m)=-q(v)m$.

Now try to abstract this as follows. The bilinear form $B(x,y)=q(x)+q(y)-q(x+y)$ determines a natural transformation $\varepsilon:TT\to\text{identity}$, where $T$ is the endofunctor $T=V\otimes-$ on vector spaces. A Clifford module structure on $M$ in these terms is a morphism $\mu:TM\to M$, and - here starts my question - certain relationship between the composite $\mu\circ T\mu:TTM\to TM\to M$, and $\varepsilon_M:TTM\to M$.

The question is what minimal structure does one need to capture this relationship. Seemingly either some kind of nonadditive transformation $\delta:T\to TT$ is needed to express $v\mapsto v\otimes v$, or some kind of self-distributive law $\text{switch}:TT\to TT$. In the latter case however one seemingly needs the additive structure to express $x\cdot(y\cdot m)+y\cdot(x\cdot m)=B(x,y)m$.

Has any of this been carried out somewhere? Is it possible to avoid the additive structure, at least using some restrictions? For example, if $B$ is nondegenerate, the endofunctor $T$ will become self-adjoint, maybe one can use this somehow, I don't know how.

As Liviu Nicolaescu points out, this probably needs some motivation. My motivation is purely abstract-nonsensical in this case. It is known that the category of modules over any algebra can be uniquely (up to equivalence) determined by an abstract category-theoretic universal property. This is because for an algebra $A$ the functor $A\otimes-$ gets a monad structure, and the category of algebras over a monad is a lax limit in the well known way.

Now for a Clifford algebra, the monad is very special, so that the category of algebras over this monad is equivalent to another category with objects determined by more concise data. I have only described part of these data, but morally this looks like (lax (left)) categorification of the fixed point set of an involution. And the question can be reformulated as follows - given a natural transformation $\varepsilon:TT\to\text{identity}$, is there a category-theoretic universal construction (some sort of lax limit again, presumably) that would yield the category of Clifford modules in the particular case when $T=V\otimes-$ and $\varepsilon$ is induced by a $q$ as above?