Johns Fish Market, Black Dog Cafe, Scottish Bakehouse, Net Result (both Vineyard Haven and Menemsha), Giordanos Pizza - outside "slice" tables, The Bite - Menemsha. Edgartown is spotty. There is one place down the alley next to David Ryans (can't remember the name) that although they have outside seating, the dogs need to be outside the fence..but it does work. There is also a "fine" restaurant on the main drag just up from the movie theaters (also the name escapes me - you can tell I don't get to Edgartown much ;) ) that has a few outside tables that you could probably have a smallish dog. All I can think of at the moment.
David Ryans Restaurant and Cafe