I am interested in sheaf theory and am slowly working towards familiarizing myself with it. This question is to further this understanding.
Let $\pi:E\rightarrow M$ be a finite-rank real vector bundle over the smooth real manifold $M$. Let $$ \Gamma_E:U\in\tau_M\mapsto\Gamma(E|_U) $$denote the sheaf of smooth sections of $E$, and $\mathcal F_M:U\mapsto C^\infty(U)$ the sheaf of smooth functions.
As far as I am aware most (or is it all?) information about the vector bundle is contained within $\Gamma_E$. Moreover, reading the wikipedia section about Étalé spaces and stalks, I get the impression that constructing the Étalé space from the stalks is very similar to constructing the total space of a fiber bundle from its fibers.
However, the stalk of the sheaf $\Gamma_E$ at $p\in M$ is not $E_p$ but instead the germ of smooth sections of $E$ at $p$. The germs clearly contain more information, as they also contain information about the bundle on all infinitesimal neighborhoods of $p$, while the fibers only contain information about the bundle at $p$.
So the questions are:
Is there any sheaf-theoretic way to construct the fiber spaces $E_p$ from the sheaf $\Gamma_E$?
How is the étalé space of $\Gamma_E$ related to the total space $E$ of the vector bundle?