  VB-groupoids define a special class of Lie groupoids which carry a compatible
linear structure. In this paper, we show that their differentiable cohomology
admits a refinement by considering the complex of cochains which are
k-homogeneous on the linear fiber. Our main result is a Van Est theorem for
such cochains. We also work out two applications to the general theory of
representations of Lie groupoids and algebroids. The case k=1 yields a Van Est
map for representations up to homotopy on 2-term graded vector bundles.
Arbitrary k-homogeneous cochains on suitable VB-groupoids lead to a novel Van
Est theorem for differential forms on Lie groupoids with values in a
representation.
