I am pretty sure if you make the wrestling team, USNA will require you to register with NCAA.
Having gone through this with my DD (registering with NCAA after she got to the Academy), I would suggest you take care of it now. The registration not only requires a $65 fee but also requires you to complete a questionnaire about the courses you have taken, send in SAT/ACT scores and submit a high school transcript which might be difficult to arrange for once you are at the Academy (you will have no computer access during the summer, your hs may have limited hours during summer and someone will need to go to hs school during summer to arrange for a transcript). Also, if you make the team, you may not be able to work out with them until NCAA has confirmed your status.