High Thyme is an upscale restaurant on Sullivan's Island. They always try to have at least one veg/vegan entree on the dinner menu, which is served Wednesday to Saturday 5 pm to 10 pm. Menus include items such as Pan Fried Lentil Cakes with Curry Potato Puree. The brunch and tapas menus are less vegan-friendly.
Mex 1 Coastal Cantina is a casual Mexican restaurant. The menu includes "shroomer" tacos and baja bowls, just ask for no cheese. Their Casa Ensalada is vegan and you can get tortilla chips with fresh guacamole or smoked chipotle salsa. And for a tasty beverage, the margaritas and cocktails at Mex 1 are all made with fresh juices.
Papi's Taqueria is a cute restaurant just steps from the ocean that sells-obviously-tacos. You can get a taco with beans or a tortilla salad (for both specify no cheese or crema) and chips with a selection of dips. They also have a nice selection of cocktails. Perfect after a day on the beach!
The Co-Op on Sullivan's Island is a small market and cafe. Their menu includes a veggie sammie (ask for no cheese), avocado toast and bagels. Their grab-and-go section usually includes some vegan options also, like veggie sushi or Thai peanut pasta. Sometimes they offer a vegan soup, also. The menu changes constantly. They also offer a frozen wine drink called a Frose that's pretty refreshing on a hot day at the beach! You can also grab snacks and beer and wine to-go from the market. They feature local products.