If I understand correctly, there are certain cases where a convert can't serve on a beis din judging a native-born Jew. Yet when we break into groups of three for hataras nedarim (annulment of vows), I've never seen anyone stop and ask, "excuse me please but are you a convert?" (Generally a good idea as this would be incredibly hurtful.) I'd strongly assume this is okay, but does anyone have a source?
The isur comes from som tasim alecha melech- all appointment that you do should be from your brethren. This refers to positions where you are forcing people into judgement. But in a "non-appointed" position where the baal din or the noder comes of his own volition, that isn't a problem. (Aruch Hashulchan C.M. 7:1)