Mathcounts may be challenging. I know that here in PA all teams are school-based and I think that’s true nationally.
AoPS has a list of math clubs, circles, etc. Getting involved in these is better, IME, than most school-based programs. Ours offers the AMC 10/12, AIME, etc.
I know that a student in our club found a school where he could take the AMC8, which is also school-based. Leaders of a math circle would probably have dealt with this issue before.
Worst case, we have 5th and 6th graders that just dive in and take the AMC10.