The Straw Market in downtown Nassau, The Bahamas, is famous. I found it to be too mass produced. If you want to support some of the independent artists and shop owners, then you will have to get in a car and go slightly (ten minutes) away from the main street. The reason for the spread out shops is that, apparently, the rent on the main strip is quite high. The list of independent shops is super short:
Bahama Hand Prints, Ernest Street: Super expensive fabric products from clothes to bags to curtains. The best part is watching the printing of the actual cloth in their attached workshop. The website is a little slow to load, but the shop is quite nice. (also, a secret tip is that they have a bathroom, if you need one).
Kim Smith The Place for Art, #20 Village Road: He offers classes, has a frame shop, and sells prints and original artwork.
Farmers Market, on the porch of the building in the same complex as the Kim Smith gallery. Basically, it’s two vendors. One who sells some vegetables, a baked good, and lobster tail tacos. The other person makes the tacos.
Bahama Art and Handicraft, East Shirley Street: Two sisters have this shop with lots of nick knacks. They don’t allow you to take photos of the things inside but you can find them on their Facebook page if you want to see what they have.
That’s it. But, if you stop at the parking lot where the fresh conch is being shucked , you can make a fun day outing of shopping local.