Closest airport is on-island, but flights are limited. You could take a taxi from there very easily.
Next closest airport: Savannah. You can catch a shuttle to HHI. Haven't priced it out lately, but believe it runs in the $30 range pp. Best bet is to call SAV and confirm.
Many people enjoy having a car in order to see all that the island has to offer. Food for thought.
I just got back from HHI and I think it would be difficult without a car. I guess if you enjoyed biking it may be okay but I think a car is almost a necessity.
There is a small airport right on HH. But the next closest airport is Savannah, Ga. It is less than an hour away. My husband and son had to fly into it 2 weeks ago, I had driven down my other 2 children (oldest son had a swim meet he needed to stay a few days back for). They took a shuttle service from Savannah airport that cost $59 for both of them in total -- very reasonable considering what they had to pay to get from our home to LaGuardia Airport. It took them right to the Disney resort. Not sure what your situation is, but you really want to have a car while you are at HH.
I can not imagine being on HHI without a car. The distances are just too large. Plus half the fun is going out and exploring. I think it would be a disappointing trip if you just restricted yourself to just the resort and what is within walking distance.