Let $G$ be a (nice enough) topological group (actually a filtered colimit of compact Lie groups), and let $X$ be a manifold with an action (a proper one in fact) by a Lie group $H$. Let $X//H := (X\times EH)/H$ be the Borel construction.
Does the following claim appear in the literature:
Claim: The set of isomorphism classes of $H$-equivariant principal $G$-bundles is given by $[X//H,BG]$
For the case when $H$ is a compact Lie group and $G$ is an abelian Lie group, then this is proved in a 1983 paper by Lashof, May and Segal,
Equivariant bundles with Abelian structure group, Contemporary Math 19 (1983) 167--176. (pdf)
And some related cases are treated by May in
Some remarks on equivariant bundles and classifying spaces, Asterisque 191 (1990) 239--263 (pdf)
The claim seems intuitively true, but I can't seem to be able to find it, so my intuition may be off. If it is true, then surely May would have mentioned it (perhaps he does and for some reason I missed it).
EDIT: May also treats the topic in chapter VII of the book Equivariant homotopy and cohomology theory, but it seems a compilation of older work.
Another way to view the question is to ask if the categories of H-equivariant G-bundles on X and ordinary G-bundles of X//H are equivalent. There is some work in Lashof-May-Segal where they refrain from assuming G is abelian (there called A, however), I'm digesting it at the moment.
Also note I'm not asking for a classifying space for equivariant bundles - this is a much different question.