  Given a bundle gerbe on a compact smooth manifold or, more generally, on a
compact \'etale Lie groupoid $M$, we show that the corresponding category of
gerbe modules, if it is non-trivial, is equivalent to the category of finitely
generated projective modules over an Azumaya algebra on $M$. This result can be
seen as an equivariant Serre-Swan theorem for twisted vector bundles.
