If I am reading the OP's post correctly, they are asking if not scholarship will they check their medical records. As I read it they are a walk on that has not done a DoDMERB exam.
Just my guess for AFROTC since they did not say which ROTC.
The answer is prior to commissioning you will be given an exam, and if you have had any medical conditions that required prescribed meds for a long time after the age of 13, i.e. not talking amoxycillin for bronchitis with no refills, you are required to inform them. at that point if there is an issue they may request your doctor's records.
I know for our DS because he went rated he was sent to Wright Pat AFB for a 3 day physica. He was required to bring hus personal medical records since birth.
l don't know what they do for non-rated, but my guess is if you are on any meds it will be found out long before you commission, probably when you get selected for SFT because if you are on meds for a condition you will need to take them with you to SFT. Any and all meds would have to be approved by them prior to attending.