From what I can tell, there are three different airstrips, nothing that could handle a large passenger capacity. It appears that Delta, Jet Blue, American Airlines and United all fly to the island. I am not sure if these are one time flights, but JetBlue appears to fly round trip twice a week. (All the flights are through Atlanta for all the carriers)
If Disney is serious on developing the island, the airstrip on the island's southern end could be transformed to support the needs of an Aulani type resort. The biggest airstrip appears to be on the north side of the island, but since the island is 110-kilometers long I would assume they would not opt for that route.
On a different note, if Disney does buy an airline as some rumors suggest, a plane could be placed on a route to the island that would travel between it and Atlanta, and possibly Orlando as well.
If this actually is true, I could see the 1-hour announcement going from announcing the airline acquisition, then suggesting packages which include the air tickets with the parks and hotel, and then announcing a Lighthouse Point expansion with the idea of connecting it with one of their new planes. Corporate synergy at its finest.
Will this actually happen? Doubt it. Purely hypothetical.