  VB-groupoids and algebroids are vector bundle objects in the categories of
Lie groupoids and Lie algebroids respectively, and they are related via the Lie
functor. VB-groupoids and algebroids play a prominent role in Poisson and
related geometries. Additionally, they can be seen as models for vector bundles
over singular spaces. In this paper we study their infinitesimal automorphisms,
i.e. vector fields on them generating a flow by diffeomorphisms preserving both
the linear and the groupoid/algebroid structures. For a special class of
VB-groupoids/algebroids coming from representations of Lie
groupoids/algebroids, we prove that infinitesimal automorphisms are the same as
multiplicative sections of a certain derivation groupoid/algebroid.
