PLUM ISLAND, N.Y.—This windswept island about the size of Central Park has the makings of a vacation destination, complete with views of Long Island Sound, and all of it a short ferry ride from Connecticut and New York.

But the future of Plum Island, home to an animal disease laboratory run by the U.S. Department of Homeland Security, is as murky as the fog that regularly envelopes the spit of land off the tip of Long Island’s North Fork.

The...