I think I read somewhere that if you cannot get a check performed, just have them print out a form that says they could not do one, and you can just submit this to West Point.
I really recommend it, but rest assured that even if you do not visit it in person it is probably one of the most beautiful places I have ever seen. You'll probably very much like it!
The processes for the nomination and an appointment are separate. If on West Point's portal it says you were rejected, I'm afraid that's it for this year. Definitely try again though!