In my strictly private this-is-not-legal-advice opinion, it is the festival's organizers, or the owners of the place where you are performing, who are usually responsible for buying ASCAP and BMI licenses. I think usually your group wouldn't itself need to license the performances. But much may depend on particular circumstances.

One possible approach: have a repertoire of public domain music ready. Then if you perform in a place or at an event which has no licenses (obviously you'll need to check beforehand), you can still sing without bringing down the thundering wrath of the copyright cops on yourself or anyone else. If they have licenses, so much the better: then you can perform anything in the catalog that they have licensed, as well as the PD stuff.

T.