One can define singular homology directly for an orbifold, mimicking the standard constructions, but I don't know if this has been written up. The alternative is to take a space that is homotopy equivalent to your orbifold, and take the homology of that. This has been worked out by Behrang Noohi in <http://arxiv.org/abs/0808.3799>.

If $G$ is a group, then the homology of the classifying orbifold of $G$ (the stack quotient of a point by the trivial action of $G$) is the group homology of $G$; so you see that it does not vanish above the dimension of the orbifold, which is 0, except in the trivial case.