Whilst many of the local restaurants will be closed on Christmas Day all the hotel (not just the Disney resorts) restuarants will be open and available to book.
We have eaten at Bomas in AKL on previous Christmas nights and this year will be booking the Christmas Buffet at Celebration Hotel once they start taking reservations.
All the restaurants in DTD will be open too.
We have found the supermarkets to be quiet in the run up to Christmas - certainly nothing at all like the mad frenzy of buying that goes on here - which is great *** that is exactly what I am escaping from. The thought of cooking a Christmas dinner in a strange kitchen is not what I go on holiday for!
Everyone is different and I'm sure you will enjoy yourselves whatever you decide to do.