Since Mike asked this question almost 10 years ago, Schwede's work on global equivariant homotopy theory, alluded to by Tyler in the comments, has become a classic. I'm personally not aware of the connection between orbifolds and equivariant homotopy theory having been deeply explored, but maybe it has (and maybe somebody will chime in on this question to talk about it!). It appears that a seminar on the intersection of the two subjects was held at Columbia in 2019. From the syllabus, I would guess that there hadn't been much work bridging the fields to that point, because the topics appear to be pretty clearly delineable between "equivariant homotopy theory" talks and "orbifold" talks.
However, there is recent work of Juran constructing a global equivariant stable homotopy type from any orbifold. This seems like an area ripe for cross-fertilization.
Something about the way both subjects are especially fond of finite groups really screams to me that they must be related.
One would think there'd be an analogy
orbifolds : global equivariant homotopy theory :: manifolds : ordinary homotopy theory