The first characters that came to mind are Sokka, Suki, Ty Lee, and Mai from Avatar:TLA, the non-benders. Sokka even got a "I'm not a bender so I feel useless" episode.

I wouldn't really count Mei Changsu for this trope... Yes, some people have basically magical martial arts skills, but e.g. the Emperor doesn't, and iirc we don't really see Xia Jiang or Xie Yu display them either; it doesn't really feel like this trope to me.