I don't know the route from the airport but there is a Goodings at Crossroads. (less than a mile away) It is more expensive than most grocery stores but if you only need a few things it may be okay.
Probably not the closest grocery store, but there is a Super WalMart on Turkey Lake Road - take Palm Parkway in a northerly direction (away from Disney) - it will turn into Turkey Lake Road. The Walmart is on the left just a little past the Comfort Suites.
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