I am currently in primary and I have an asthma waiver. I am only required to keep an inhaler on me at all times when I fly (not that I need or use it). It is entirely dependent on how severe your symptoms, if you have any, are. Your general service waiver will be handled at MEPS. They handled my waiver process after I graduated from OCS, and they sent me down to Walter Reed Hospital in Washington D.C. where they did a full battery of cardio-pulmonary tests, then sent it to NAMI.
Edit: After graduating OCS, I was kept in Student Pool there for a few months for the waiver process. NAMI takes time, unfortunately.