Although Wikipedia will tell you Middleton Island has an airport, don’t be fooled. The so-called airport is really just a landing strip with no facilities and no flights open to the public. If you want to visit Middleton, it should be possible to charter a plane or boat to take you there, but be prepared for a long stay. Middleton is remote with no permanent residents. There is an FAA weather station on the island that usually has a small crew and seasonally there are biologists at the USGS field camp but nowhere to buy food, supplies or lodging. I have heard rumors that there is (or was) a company that lead guided birding tours to the island but I haven’t been able to find it.
Weather is also a major consideration, Middleton is cloudy most of the time and planes in and out of the island are often delayed for days to weeks, so if you visit, it’s important to have at least supplies for a week longer than your intended stay. Dress warm with plenty of layers and a raincoat!