Given vector bundles $E$ and $F$ over a smooth manifold $M$ do we have an isomorphism $$\Gamma(\text{Hom}(E,F)) \cong \text{Hom}_{C^\infty(M)}(E,F)?$$
That is, if I have a smooth bundle map $E \to F$ do I obtain a section of the bundle $\text{Hom}(E,F)$ and vice versa? I see this being used in multiple places, but I have not found a proof for this proposition.
I am not completely sure where your problem lies and what you mean by the notation $\text{Hom}_{C^\infty(M)}(E,F)$. Sections of the bundle $\text{Hom}(E,F)$ are in bijective correspondence with vector bundle homomorphisms $E\to F$ whose base map is the identity on $M$. If $\Phi:E\to F$ is such a homomorphism than for each $x\in M$, $\Phi$ maps the fiber $E_x$ of $E$ at $x$ to the fiber $F_x$ of $F$ at $x$ and the restriction $E_x\to F_x$ of $\Phi$ bey definition is linear. Hence for each $x\in M$, we obtain a linear map $s(x)\in \text{Hom}(E_x,F_x)$ and the latter space is exactly the fiber of the vector bundle $\text{Hom}(E,F)$ at $x\in M$. So $\Phi$ gives rise to a map $s:M\to \text{Hom}(E,F)$ which maps each $x\in M$ to an element of the fiber over $x$.
Conversely, a section $s$ of $\text{Hom}(E,F)$ associates to each point $x\in M$ a linear map $s(x)\in \text{Hom}(E_x,F_x)$ and you can just peace those together to define a map $\Phi:E\to F$ which covers the identity on $M$. So you obtain a correspondence in both directions and the only question is whether smoothness of $\Phi$ is equivalent to smoothness of $s$. But this is easy to see directly looking at local trivializations of $E$ and $F$ (over the same open set $U\subset M$) and the induced local trivialization of $\text{Hom}(E,F)$.
If by the notation $\text{Hom}_{C^\infty(M)}(E,F)$ you mean maps $\Gamma(E)\to\Gamma(F)$ that are linear over smooth functions on $M$, then you only need the additional observation that such operators are equivalent to vector bundle homomrophisms that cover the identity on $M$. This is similar to the characterization of tensor fields as operators acting on one-forms and vector fields. It mainly amounts to proving that if $\Psi$ is such an operation, then for $s\in\Gamma(E)$ and $x\in M$, $\Psi(s)(x)$ depends only on $s(x)$.