$\DeclareMathOperator\Vec{Vec}\newcommand\Sch{\mathrm{Sch}}\DeclareMathOperator\Hom{Hom}$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.



"Quasi-coherent" vector spaces in Sch/S
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 $\Omega'\_{V,S}$.


