I believe that the latest seating is around 9:30 pm based on experience, but I don't know the exact time. We have dined there several times after IllumiNations.
You can go to the Dining Reservations page and it will tell you the hours that the restaurant is open. If you put in the reservation info it will show you what is available. The last res is probably around 9:45.
It currently says Flying Fish is open 5:30 - 10:00.