According to this page Ellis Island is open on Thanksgiving Day. But what about the ferries? If they don't run on Thanksgiving Day then even if the island itself is open, visiting it is not going to be very feasible. The ferry website doesn't have their schedule for Fall up either.
The ferry, Liberty Cruises doesn't have the schedule published for the month of November, however, you can reserve tickets for November 26th, which is this year's Thanksgiving day.
And both islands: Ellis and Liberty are open every day except Christmas, so it would stand to reason that if Liberty Island cannot be reached without a ferry the ferry must be running (even though it doesn't officially say so yet).