In algebraic and analytic geometry, vector bundles are usually interpreted as locally free sheaves of modules (over the structure sheaves). They are in particular examples of quasi-coherent sheaves. If the bundle is of finite rank, then the sheaf is actually coherent and this is good for certain cohomology groups to have finite rank for example.
I think the equivalence of vector bundles and locally free sheaves holds as well for the categories of topological spaces and smooth manifolds, and locally free sheaves are in particular quasi-coherent. The question is, when are they coherent? maybe it is best to ask first about the structure sheaf itself. In algebraic geometry the structure sheaf is coherent for neotherian schemes and in analytic geometry the structure sheaf of a complex manifold is coherent. What about smooth manifolds for example?