Anyone can make a reservation for a WDW restaurant - dining plan or no dining plan, onsite or offsite, tourist or local.
To clear up the CRT thing, they require prepayment in full at the time of booking. They are the only signature restaurant that does this, and the only other meals you need to do that for are dinner shows. If you are on the dining plan at the time you book, or you later switch to the dining plan and have them refund your payment, you must leave a credit card number to guarantee your reservation. They will charge if you don't cancel or you don't show up.
Chef Mickey's has no such requirements.