Here's a random result which I suspect is true: take a proper, cocompact Lie groupoid. So all the automorphism groups of objects are compact Lie groups, and the (topological) space of isomorphism classes is compact. Then there should be a presheaf on the groupoid (that is, on the manifold of objects, and equivariant for the groupoid action) that assigns the appropriate equivariant/groupoid vector bundle K-theory to an open (there's details about compactness here: perhaps one needs to be tricky with the site, and use local compactness etc, or perhaps used compact regular closed subsets or something). And I think this should be related to a constructible sheaf either on the groupoid like this, or on the quotient space, using the nice stratification that comes from work of Weinstein, Crainic–Struchiner, Pflaum–Posthuma–Tang on local linearisation. The stalk at an object (or iso class) should be representation rings of the automorphism group. If the automorphism group is connected, I could imagine that the conjugation action inducing isomorphisms on the representation rings is such that you have something on the quotient space, without that I'm not so sure. It might be ok. I had vague ideas that I wanted to use this for looking at a groupoid analogue of the result about the Brauer group compared to torsion classes in cohomology, but it's been a while, and I can't fully reconstruct the link. I have other ideas that are separate from this, and which feel more promising, but for my sins, I have to write up the proof for compact manifolds first.