Apparently they're much more locked into a particular institution than when I got mine. As I remember from (holy crap) 16 years ago, I could go to any of the 3 schools I listed on my NROTC app. Now they apparently award the scholarship only to a particular school. You are restricted to schools with an NROTC unit or a crosstown affiliation, for sure, though. I don't know about Stanford, but the only Ivy League schools with NROTC are Penn and Cornell. Go Quakers!!! It's good to get a degree from a good school if you get out sometime, but honestly, the Corps doesn't care whether you got your degree from USNA, Penn, or the Rocko School of Typewriter Maintenance.
Anyway, are you sitting on offers for NAPS and Marine-Option NROTC right now?
First, the NROTC will see you graduating with a commission in 4 years instead of 5 with NAPS.
Second, you can still reapply to the academy from a civilian college. This is what I had planned to do when I went to school. Then I realized this college gig wasn't half bad, and I already had 1 year down.
The Academy v. NROTC debate will never end. There are several threads on the topic. In your circumstances, though, take the bird in the hand and go NROTC. The Academy may be good, but it's not worth waiting another year if you already have an alternative offer.