I am an adult skater but I think the answer will be the same.
I am a member of my longtime club but moved away from the area. I am not a member of my new area's club. Both of the club's are members of US Figure Skating.
I skated and competed with the Theatre on Ice team in my new area. I did not need any "permission" to skate with them, just the membership in good standing with US Figure Skating.
But really, if you have concerns you should check with your club board members. There will be times when membership is crossed-checked with US Figure Skating and your home club and a board member will validate that you are in good-standing.