The Turks and Caicos aren’t known for their shopping, although there is duty-free shopping here. There are several main shopping areas on Provo: the Ports of Call shopping village (probably the best variety), the Saltmills complex, and La Petite Place in Grace Bay, and Market Place and Central Square on Leeward Highway (about ½ mile to 1 mile east of downtown).
Look for hand-woven straw baskets and hats, polished conch-shell crafts, model sailboats, and handmade dolls among the native crafts. Of course, there are also plenty of souvenir t-shirts and swimwear. Rare conch pearls may be available, but are discouraged due to over-fishing.
As far as stocking your villa with food, the Graceway IGA is probably the best grocery store in Provo, and most like the supermarkets you’re used to at home.