  We study vector bundles over Lie groupoids, known as VB-groupoids, and their
induced geometric objects over differentiable stacks. We establish a
fundamental theorem that characterizes VB-Morita maps in terms of fiber and
basic data, and use it to prove the Morita invariance of VB-cohomology, with
implications to deformation cohomology of Lie groupoids and of classic
geometries. We discuss applications of our theory to Poisson geometry,
providing a new insight over Marsden-Weinstein reduction and the integration of
Dirac structures. We conclude by proving that the derived category of
VB-groupoids is a Morita invariant, which leads to a notion of VB-stacks, and
solves (an instance of) an open question on representations up to homotopy.
