We consider how the geometry and topology of a compact $n$-dimensional Riemannian orbifold with boundary relates to its Steklov spectrum. In two dimensions, motivated by work of A. Girouard, L. Parnovski, I. Polterovich, and D. Sher in the manifold setting, we compute the precise asymptotics of the Steklov spectrum in terms of only boundary data. As a consequence, we prove that the Steklov spectrum detects the presence and number of orbifold singularities on the boundary of an orbisurface and it detects the number each of smooth and singular boundary components. Moreover, we find that the Steklov spectrum also determines the lengths of the boundary components modulo an equivalence relation, and we show by examples that this result is the best possible. We construct various examples of Steklov isospectral Riemannian orbifolds which demonstrate that these two-dimensional results do not extend to higher dimensions.