Here is what I remember about this topic:
1. For athletics eligibility in general, I recently read that the NCAA is extending the age limits for athletes to make sure prior military can play. (This had always been the case, then the NCAA apparently accidently closed the military loophole, and now they are re-opening it.)
2. If you're a mid at USNA, you're eligible for all teams.
3. I believe Navy football does have one walk-on tryout day each year. Generally they take no one. Navy football tends to over-recruit because they know they'll get drop-outs from NAPs and academics and the demands of plebe year, etc. They also can effectively offer EVERYONE a "scholarship," in the sense that all mids go to school for free in return for the military obligation, so there's nothing to stop them from trying to bring in as many football recruits as admissions will allow.
4. Without high school football experience, your chance of making Navy football as a walk-on is, I would say, functionally non-existent. If you end up in the Marines, you can try to see if there is a base team on which you can play and you can try to get a friend to take film of you if you are good. To have even the remotest chance of walking on, you'd really need to make prior contact with coaches and have film to interest them.
5. Sprint football at Navy (aka "lightweight football") is also quite competitive and generally is made up of very successful high school players (all-league, all-conference), but you could check it out.
6. If you have your heart set on playing a sport under the "N star" you should start thinking widely about things like rugby, as well, to give yourself some options.
7. There is nothing wrong with being motivated to play a sport at the Academy, but you've got to get in first. Enlistment as a Marine can lead to USNA but it is not certain, so just make sure you are okay with serving your full hitch in the USMC (there are other office routes too) if USNA doesn't work out.