From the basics of sheaf theory I know that we have an equivalence between sheaves on a topological space $B$ and certain spaces $\pi:E \to B$ over $B$. This is equivalence is given on one hand by the functor \begin{equation} E \mapsto (U \mapsto \Gamma(U, E)) \end{equation} its quasi-inverse is given by defining a topology on $\coprod_{b \in B}F_b$, where $F$ is a sheaf of sets.
Now I'm trying to understand if there is a way to do something like this between sheaves of $\mathbb{F}$-vector spaces (where $\mathbb{F} = \mathbb{R}, \mathbb{C}$) and (not necessarily locally trivial) Bundles of vector spaces.
That is, consider $\mathbb{F}$-$\mathrm{Bun}_B$ the category of spaces over $B$ endowed with a vector space structure on its fibers (such that the operations are continuous in the usual manner) and consider $\mathrm{Sh}_\mathbb{F}(B)$ the category of sheaves of $\mathbb{F}$-vector spaces over $B$.
There is a functor $\Gamma: \mathbb{F}$-$\mathrm{Bun}_B \to \mathrm{Sh}_\mathbb{F}(B)$ because of the continuity of the operations just like in the case of sets.
Now to construct a quasi-inverse $\operatorname{Et}: \mathrm{Sh}_\mathbb{F}(B) \to \mathbb{F}$-$\mathrm{Bun}_B$ I guess I would want to define a topology on the space $\coprod_{b \in B}F_b$ making the canonical projection $\operatorname{Et}(F) = \coprod_{b \in B}F_b \to B$ into a (NNLT) $\mathbb{F}$-bundle.
The usual way is to define a topology making each section defined by $F(U)$ continuous, but this won't work: if $E \to B$ is the trivial bundle $\mathbb{R} \to \{*\}$, then $\operatorname{Et}(\Gamma(F))$ is $\mathbb{R} \to \{*\}$, where $\mathbb{R}$ has the discrete topology (ewwww).
My question is: Is there any way to do this construction and remember the $\mathbb{F}$'s topology so this construction defines an equivalence? Is an equivalence between these two categories even reasonable to expect in the first place?
A little motivation: All of this questions have arisen because I was trying to make some proofs of vector bundles simpler. For example if there is such an equivalence then maybe we can define the tensor produt and the direct sum of bundles in the world of sheaves! Of course, it's not haaaaaard to do these things in the topological world, but who knows?