I applied for the Army and AF concurrently. The Army set up my MEPS since they were the first ones I started. About a month, later the AF was able to get my physical from the MEPS station for their packet. The SF form (standard form) used for security clearances is the same accross the services.
One word of advice though. As a courtesy, please let both services know that you are applying for two scholarships if you choose to pursue both concurrently (to have one in the event you don't get the other).