Folks . . .
PLEASE PAY ATTENTION TO THIS.
First, if you have a medical condition, and need medication for it (whether an inhaler or anything else), do consider whether it's in your best interest to stop that medication. Have a SERIOUS talk with your doctor. The last thing you want to do, in the interests of attending a SA or ROTC, is to create a serious medical condition for yourself.
Second, to the best of my knowledge, you cannot ask a doctor to change your medical records. The one exception might be if, upon reading them, you discover an error. The doctor could add a note explaining the error; that's different than removing something.
Third, DODMERB asks about your medical history. Lying is definitely not how you want to start your military career. You need to be honest about conditions you have. Thus, regardless of what your medical records say, if you've had a medical issue that DODMERB asks about, you need to disclose it.
Fourth, DODMERB disqualifies (and SAs/ROTC choose not to waive) for good reason. As has been detailed in the past, if you are stupid enough to get by with an undisclosed medical condition, you present a huge potential liability to yourself and your comrades. IOW, you could get yourself killed and them killed trying to help you.
This is especially true with asthma. An asthma attack in the desert, on a mission, etc. could create horrid repercussions for everyone.
I'm NOT a medical expert, but there are various tests, etc. that will help determine whether you do have asthma. If you do, your chances of getting a wavier are very small. If you don't -- you SHOULD be okay even if you were prescribed an inhaler after age 13.
Fifth, getting rid of your inhaler now won't make much difference as I believe the question is whether or now you have had symptoms or been prescribed medication for asthma after age 13. If you've carried one around for the past 4 yrs and used/never used it, it won't matter whether you still carry one around today -- process will be the same.