Here is a recent thread with satellite photos/directions to grocery stores. To answer your direct question, probably the Winn Dixie on Apopka-Vineland is the closest "reasonable" grocery store (the Goodings outside DTD is closer but way over-priced). It's maybe 10-12 minutes from the check-in area of the Fort. But there are several to choose from. If you're a Publix shopper at home, for example, there's several that are not much further.
Walmart, Winn Dixie, Publix are all within a 15 minute drive from FW. We usually stock up for our trip so we do not have to do any major shopping once we get there. But with a 2 week stay, there is always at least a mid trip run to get some perishibles restocked.