$\def\VB{\mathsf{VB}} \def\sO{\mathcal{O}} \def\Mod{\mathsf{Mod}} \def\LFMod{\mathsf{LFMod}}$For a ringed space $(X,\sO_X)$ and a ring $R$, denote $\Mod(\sO_X)$ and $\Mod(R)$ to the categories of $\sO_X$-modules and of $R$-modules. Let $M$ be a smooth manifold and denote $\sO_M$ to the sheaf of smooth functions on $M$. Denote $\VB(M)$ to the category of smooth vector bundles over $M$ with vector bundle homomorphisms. For a vector bundle $\pi:E\to M$ over $M$, denote $\Sigma_E$ to the sheaf of sections of $\pi$. There is a functor \begin{align*} \Sigma:\VB(M)&\to\Mod(\sO_M)\\ E&\mapsto\Sigma_E\\ F:E\to E'&\mapsto F_*:\Sigma_E\to\Sigma_{E'}, \end{align*} where $F_*$ is induced by postcomposition by $F$. (Actually, postcomposition induces a morphism of hom-sheaves $\mathcal{H}om_{\VB(M)}(E,E')\to\mathcal{H}om_{\sO_M}(\Sigma_E,\Sigma_{E'})$). Working in a local frame, one can see that $\Sigma$ is faithful by seeing that $\Gamma\circ\Sigma$ is faithful, where $\Gamma:\Mod(\sO_M)\to\Mod(\underbrace{\Gamma(M,\sO_M)}_{C^\infty(M)})$ denotes the global sections functor. Less trivial is that the functor $\Sigma$ is also full. For this, one first checks that the functor $\Gamma\circ \Sigma$ is full: this is Lemma 10.29 of Lee's Introduction to Smooth Manifolds. Leveraging Lee's proof, one can show after fullness of $\Sigma$: Given vector bundles $E$ and $E'$ over $M$ and an $\sO_M$-linear map $\varphi:\Sigma_E\to\Sigma_{E'}$, denote $F^U:E|_U\to E'|_U$ to the unique vector bundle homomorphism such that $\Gamma\circ\Sigma(F^U)=\Gamma(F^U_*)=\varphi_U$. Using the explicit definition of $F^U$ given in Lee's proof of Lemma 10.29, one can check that $F^U|_{U\cap V}=F^V|_{U\cap V}:E|_{U\cap V}\to E'|_{U\cap V}$. Therefore, the $F^U$'s glue to a vector bundle homomorphism $F:E\to E'$ such that $F|_U=F^U$. Thus, since $F_*|_U=(F|_U)_*=F_*^U$, one has $F_{*,U}=\Gamma(F^U_*)=\varphi_U$, i.e., $F_*=\varphi$.
This shows that $\Sigma$ is a fully faithful functor and hence an equivalence of categories into its essential image. It is clear that the image of $\Sigma$ is contained in the full subcategory of locally free $\sO_M$-modules. What I am interested is on finding a detailed proof that locally free $\sO_M$-modules are in fact the essential image of $\Sigma$. The only reference I am aware of is Ramanan's Global Calculus. He discusses this essential surjectivity in Chapter 2, between definitions 2.8 and 2.9. However, I find the argument sketched on his discussion a little bit unsatisfactory.
So my questions are:
Do you know any reference which proves the essential surjectivity of $\Sigma:\VB(M)\to\LFMod(\sO_M)$ with more detail than Ramanan does?
Do you know yourself a more detailed proof of this fact than Ramanan's one?
I was trying to combine the Vector Bundle Chart Lemma of Lee's book (Lemma 10.6 of the 2nd edition) with Ramanan sketch, but I wasn't sure how to actually do it.