A local quotient stack is a stack with a suitable atlas by quotient stacks.

For instance (FHT, I, A.2.2).

Every local quotient stack admits a locally universal Hilbert bundle?. Hence local quotient groupoids have a good topological K-theory. See at groupoid K-theory for more on this.

(FHT, I, lemma 3.12)

In

the groupoid K-theory of local quotient stacks is discussed.