Yes you will need to rent a car to get from the airport to the island and around the island, although Disney does have a shuttle from their resort to their Beach House in Palmetto Dunes.
Baggage claim/car rental are very easy in Savannah, the airport is only 15 gates and usually by the time we get to baggage claim our luggage is waiting for us then the rental car and we are on our way most times in less than 1/2 hour and 45 minutes later we are on the island
Hope this helps and enjoy your stay.
There is an option to fly onto Hilton Head Island rather than Savannah. To do this, you must take American Airlines and connect through Charlotte. It is a small airport so does not have as many flight time options and can be more expensive. But for some the convenience is worth it. They have rental cars on sight, but reserve in advance.
We have been flying Jetblue from New York JFK into Savannah/Hilton Head International and they have been simple straight-forward flights. I took a look at the Jetblue website, they have flights from Burlington VT but do have a layover at JFK for the connection. The Savannah/HHI airport is wonderful and one of my favorite as it is not a big terminal. Car rental counters and the actual cars themselves are onsite to the left and right of the terminal so it is quick and easy once you have landed. From there it's a 45-50 minutes drive to HHI. While there are shuttle buses from the airport to the island, we have found having a car there is a nice convenience. You will need it if you plan to visit Pickney Preserve, Harbor Town, South Beach, Coligny Plaza and some of the fine restaurants on the Island. It is a must for day trips to Savannah, Bluffton and Beaufort.Edited: 16 October 2017, 20:56
You have two options on airports: 1) the one located directly on H. Head (but with less flexibility on times & prices) or 2) Savannah, GA which is approx. 45-60 minutes away. Savannah generally has better prices but check both anyway as you may get lucky. You will more than likely want to visit Savannah while you are in the area anyway so...As with travel anywhere, time generally equates to money. Good luck!