It is well known, by Serre-Swan theorem, that given a compact manifold M there is an equivalence of categories between the category of vector bundles over M and the category of finitely generated projective modules over the space of smooth functions over M.
Question: Is there an analogous result for the category of double vector bundles over a compact manifold M? Is there a purely algebraic description of a double vector bundle?
Roughly speaking a double vector bundle is a "vector bundle object" in the category of vector bundle. This statement is no precise and the concrete definition can be find in chapter 9 of Mackenzie's book "general theory of Lie groupoids and Lie algebroids" or in the paper K. Konieczna, P. Urbanski, Double vector bundles and duality, arXiv:dg-ga/9710014v1. I don't write the definition in full detail because it is a little intrincate but we sketch it as follows:
A double vector bundle is a arrangement of manifolds $(\mathcal{B}; \mathcal{V}, \mathcal{H}; \mathcal{P})$ such that $\mathcal{B} \to \mathcal{V}$ , $\mathcal{B} \to \mathcal{H}$ , $\mathcal{V}\to \mathcal{P}$ and $\mathcal{H} \to \mathcal{P}$ are vector bundles, and such that structural maps of $\mathcal{B} \to \mathcal{V}$ are vector bundles morphisms over the respective structural maps of $\mathcal{H} \to \mathcal{P}$ from $\mathcal{B} \to \mathcal{H}$ to $\mathcal{V}\to \mathcal{P}$. The same must be hold for the other structure of vector bundle over $\mathcal{B}$.
See the cited references for a more clear (and pictorical) definition.
