The original proof of Helly's theorem was topological and only uses basic homological properties of convex sets.  It generalizes to all sorts of contexts, including the one you are interested in.  A discussion of this with references and specific statements is Section 3 of

B. Farb,
Group actions and Helly's theorem,
Adv. Math. 222 (2009), no. 5, 1574–1588.