As I recall, most grocery stores shut down. Publix definitely closes, I am almost certain Wal-Mart does as well. Winn Dixie might be open with limited hours, but I would not count on it. For basic food needs, the 24-hour Walgreens should all be open, as well as 7-11s and other mini-marts.
A surprising number of restaurants are open. Everything at Downtown Disney, but also a fair number in the surrounding area. Be warned, crowds are huge. For example, I have heard TGI Fridays, of all places, has a line at 10:00 a.m. Christmas morning. Opening seem kind of hit or miss, if you have a place in mind you might want to call in advance.