Wilson's Variety of Cockburn Harbour, South Caicos.
South Caicos isn't a major tourist destination and as such has little in the way of tourist souvernir shopping.
There are numerous local convenience and grocery stores on the island, mostly in the Cockburn Harbour area.
If you're planning on spending more than a day here, you may want to buy any specialized products from Providenciales. There's no problem finding groceries, but the selection can be a bit limited beyond that.