Farm Neck is a very good course. Although not terribly architecturally interesting, it is the "showcase" public track on the island.
Mink Meadows is a Wayne Stiles 9 with multiple tees for two times around. There even is a hole with two green complexes for the two trips around. It's undulating, fun, and reminds of Plymouth CC on the mainland.
The Vineyard Club is a very good Donald Steel course, but private.
Edgartown GC is a 9 holer that's over 100 years old. It has a small membership, and is difficult to get on, harder probably than the difficult-to-get-on Vineyard Club.
Sorry Randy, but anything not on the Vineyard itself is not "near" the Vineyard. The only think "near" the Vineyard is Chappaquiddick and dark-green cold H2O.