As you can see in the page I linked in my previous post, there are both 公設 "public" and 私設 "private" 放課後児童クラブ. ちゃんと確認してから回答してますよ。何せ母語なんですから。
Also, sorry to point out this, but "息子のおばあさんは学度を働きます" doesn't make sense at all. You wanted to say 息子のおばあさんは学童(保育)で働いています, by any chance?
Anyway, then, the most common way of saying is 学童保育を申し込みたいのですが.