「등록[登錄]하다」tends to have a meaning to "register" or "logon" somewhere, while「신청[申請]하다」generally means to "apply" for something. Thus, in my opinion, When can I register for this class? --> (나는/저는) 언제 그 과정을 등록할 수 {있습니까 / 있어요}? === 좋은 하루를 지내세요.／I wish you a good day.