I have seen people with dogs at The Boarding House, Cioppino's, and Languedoc patios. Might be ok at The Summer House Beach Bar restaurant. And maybe Topper's @ Wauwinet (on the deck where they serve lunch).
Nantucket is very dog friendly-- we always bring our lab. He was welcome last summer at both Queequegs for cocktails/apps and at Something Natural for sandwiches/salads. Shopping is easy to do on near HyLine ferry as many shops have open door policies.