Goodings is the closest to FW, but my favorite is the Florida chain Publix Supermarkets. If possible stop on your way in as to go back out can be quite a drive. If necessary stop again if you make any side trips off Disney property. There are many other stores in the local area including Winn Dixie and Goodings (a bit high in prices). The closest Publix I have found is Publix Regency Village Shopping Center.
Driving Directions:
Exit the Disney complex on Hotel Plaza Blvd (Lake Buena Vista) to 535, turning right on 535.
Drive under I-4, turning left at the first light after you cross under I-4. This should be Vineland Ave.
Continue on Vineland Ave, past the outlet stores (nice center) and follow the signs to Publix.
This is about 5 minutes further than Goodings, but you save a bunch!
Current Store Hours:
Sun 8:00 AM - 10:00 PM
Mon 8:00 AM - 10:00 PM
Tues 8:00 AM - 10:00 PM
Wed 8:00 AM - 10:00 PM
Thurs 8:00 AM - 10:00 PM
Fri 8:00 AM - 10:00 PM
Sat 8:00 AM - 10:00 PM