Hi,
I'm not sure if this is the closest but it is the one we found and visited during our visit in January.
There is a Publix Supermarket on Vineland Rd., about 3 miles south of the hotel.
Quick directions - Exit the hotel on the gateway exit, turn right to the stoplight, turn right and follow this road almost to Irlo Bronson Memorial Highway. The Publix is on your left right near the corner. This a major intersection so keep alert for the turn to the shopping area. I think it is about 3 miles.
They staff was helpful and friendly when we were there. Hope this is helpful.
Julie