If the base $X$ is compact Hausdorff, then the topological Serre-Swan theorem asserts that the category of vector bundles over $X$ is equivalent to the category of finitely-generated projective modules over $C(X)$. This naturally sits in the abelian category of all modules over $C(X)$ as an additive subcategory.
In general the category of vector bundles over $X$ should sit naturally in the abelian category of sheaves of $\mathcal{O}_X$-modules on $X$ as an additive subcategory (or something like that; I don't know much about sheaf theory).