Place down your general health care provider...i.e. ABC Pediatrics.
You don't need to have the exact doc, but if you were going to a doc for 4 yrs due to asthma, you should be able to recall the Practice's name. If you can't just place it in your history.
You were diagnosed with "asthma" and that is what you are to put into the questionnaire. There is one thing to say I don't know when you don't know because it was a 1x condition that only happened 1x, but you had it for 4 yrs. That is called a long term condition.
The fact is you need to be honest at the get go, because trying to hide the condition will come back and bite you in the arse later on. This will not be your last DoDMERB exam, as long as you are in the military you will be examined to their regs. It showing up later on will just mean further havoc since if it is a DQ issue, they can still cut you loose or place you throw the waiver process.
See this thread.
http://www.serviceacademyforums.com/showthread.php?t=21586 This is a cadet with 2 yrs in AFROTC.
Additionally, I don't know if you have a plan B, assuming you are doing this for an SA as plan A. Understand that the exam is only valid for 2 yrs., and if you get a 3 yr scholarship, most likely this exam will not be current for contracting 2 yrs from now, which means you have to do this all over again.
Finally, as I stated this will not be the last, prior to commissioning they will do a more in depth exam if you are ROTC, specifically AFROTC. It is a 3 day exam done at WP. Again, they will find it, and now your career is riding on that waiver.
Tell them the truth now, be prepared to provide documentation later on IF it is an issue. No playing ostrich in the sand here.