You can get the dining plan.
However, you have to book a room/tickets package to get it. This means you have to buy some form of ticket. AP holders sometimes book the package deal and get a one-day base ticket (every member of the party on the reservation must book the same type of ticket). You can either use the one-day ticket at a park, or use its value toward a ticket upgrade, or toward the renewal of your annual pass if you renew it.
You will also have to forego any AP discounts that may be available on your room. You cannot book a room-only reservation with an AP discount and add the dining plan. And if you have your heart set on the dining plan, it may not be wise to take advantage of the DDE program, which can save you a bunch on table service meals, plus the discount includes alcoholic beverages.
You could also rent points from a DVC member and sign up for the dining plan that way. DVC members staying on points at DVC resorts do not have to purchase tickets in order to add the dining plan.