I know that there is frequently a group at Indiana Gun Club on Wednesday evenings. I think it is affiliated with Hamilton County 4H, but I'm not 100% sure. I've stopped and talked with the adult leads before. A great group of kids. I always enjoy seeing young ones on the clay fields and am always sure to thank the adults that have the patience to take them out. I'm sure if you call Indiana Gun Club (317-485-6540), they can let you know.
Five points conservation club has a youth trap group/team near five points and Arlington on the south east side. They shoot Tuesday nights I believe for youth then the regular shoots are on Thursday nights.