Will you have a rental car? Or are you driving down?
There is Goodings just across the street from Downtown Disney. Just head down Hotel Plaza Blvd. Cross over SR 535 and it is in the Crossroads Shopping Center. The store is open 24 hours/7 days a week.
There is a PUBLIX located on Vineland. From Downtown Disney, turn right onto SR535 and go under the I 4 overpass. Get in the left lane as you will be turning left onto Vineland (there will be a Bahama Breeze on this corner) Head down Vineland and you will pass the Premium Outlets on your right (great shopping here) go around the curve and PUBLIX will be on your left just past the Dixie Stampede and the Outback Steakhouse.
If you need a Walgreens, there are 3 very close by as well. 2 are located on SR 535 (one in either direction when you come down Hotel Plaza Blvd and SR 535. One down about a mile on the left and the other if you would have turned right down on the right side just before Bennigans Restaurant and the Sheraton Vistana.
If you want
WalMart, continue down SR535 and cross over 536. About a mile and a half down on the right side there is a fairly new WalMart that is open 24 hours located on the corner of Polynesian Isle and 535.
There are also HESS stations on Property that will also carry milk and Orange Juice, bread and such.