It depends a little on where on STT are they staying. There are several decent stores around the island and several neat little ones. I found one, the Mandahl Market . while staying at Magens Point Resort. It's on Rt 39 just off Rt 42 not far from Magens Point . Great little market with small deli, take out food, booze, wine and beer. Fresh breakfast items in the AM. Just got to hunt these places out. Anyway there are markets/grocery stores on both ends of the Island. There are at least two Pueblo Markets, one on Rt 30 out near the Airport area and the other on Rt 313 near Havensight. There is the Marina Market in Red Hook and another larger store on the same road, Rt. 38, about a mile back toward Charlotte Amalie. Some people prefer K-Mart on Rt 38, Tutu area, for booze and wine. Hope this helps.

I think Food Center is the name of the one Bobcside means near the east end. Very big & very clean, stateside-side style market on the way to Red Hook. There's another very expensive but very nice market in Havensight called Gourmet Gallery. Cost-U-Less is a big box type store with a very good meat selection.

On STJ:
Starfish Market in Cruz Bay, carries a good selection similar to smaller state-side grocery stores, has bakery products and a deli. This is the major grocery store on STJ.
Gourmet Market on Salt Pond Road out of Coral Bay is a small store, limited selection, but carries the basics plus a beers and wines.