Not sure if it's essentially* the same as the already given answer(s), but a useful definition of orbifolds and morphisms thereof (that works also in other categories) is via stacks.
I think in this framework a (topological) orbifold would be a stack on the category of (not necessarily compact) topological manifolds that is locally isomorphic to a quotient stack of the form $[\mathbb{R}^n/G]$ with $G$ a finite group (perhaps embeddable in $O(n)$ for the same $n$, if you want). Note that the notion of sheaf and the analogous notion of "vector bundle" is very naturally available in this framework. There is also a theory of differentiable stacks and in particular differentiable orbifolds, where you can talk about tangent "bundles" and metrics.
${}^*$ Because a groupoid internal to a category of "spaces" is some kind of "presentation" for a stack on the same category, and Morita equivalence corresponds to the right notion of "isomorphism" (equivalence) of stacks.