The trading post is the closest, as it is inside FW
The closest "real" grocery store is Goodings which is just across the street from Disney property.
The meadows trading post has a small grocery section for emergency shopping. While the prices are not totally outrageous, they are higher than local supermarkets. If possible I would suggest stopping at a local store (my favorite is the Florida chain Publix Supermarkets) 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, pass the outlet stores (nice center) and follow the signs to Publix.
This is about 5 minutes further than Goodings, but you save a bunch!
Most companies that deliver have gone out of business, but I think there may be one or two left. Hopefully other will post names.