We have cruised in the Bahamas for the past six years. During that time, the availability of produce and provisions has improved. The following information will only be helpful to you if you are not
insistent on organically grown.
Nassau: The "box" deals on Potters Cay might not be available, but there is a local market along the street which sells fresh vegetables, conch, and some fish
(haven't bought fish
there). There is a large grocery store near Nassau Yacht Haven and Nassau Harbour Club, which is affiliated with Winn Dixie in the US. Prices used to be double US, but they have moderated somewhat and produce is good there. There are numerous other small markets in the area - last year there was an open market near Montague Bay, and one in the parking lot of the Green Parrot restaurant some Saturdays (this is very near one of the main Nassau anchorages).
Down Island from Nassau, there are small stores in many of the communities (Staniel Cay, Black Point, etc.) and the best day to go is right after the mail boat unloads its weekly supplies. If you ask around, there might be locals selling a little produce as well.
On Great Exuma, there is good produce available from a local grower at Barreterre, on the north end of the island not far from Rat Cay (it's shallow going in there, and you should get info on the radio
as to availability first). In George Town on Great Exuma there is a well stocked grocery store. During the cruising season, they even stock upscale produce, such as endive and enoki mushrooms (go figure!) There is also a small produce stand near the straw market which many cruisers frequent.
There are two good markets on Long Island
(and if you're not a vegan and have had bad fishing
luck, a good place to buy well frozen local seafood).
The upshot is - if you make landfall ask around about a local store and if there is something available, buy it. Be prepared to pay a little more, and to experiment
with various types of local produce.
And lastly, before you leave home stock up on the produce that lasts a long time - cabbage will last unrefrigerated for several months - just store in a fairly cool spot peel off the outer leaves before using. Onions, apples, etc.
Good luck and enjoy your cruising!