Everyone says that you should apply as early as possible (end of junior year). This is true for one major reason (I'm telling you this as an incentive). These scholarships are picked by a board that starts out with a certain number of them and a certain number of slots at each school. If you apply early and get the scholarship, it is more likely that you will get it (apps that are not accepted at first are looked at by the board each time it meets in case they can be accepted later; the ones left at the end are denied). Also, some schools have a very few slots and if you apply late, you may force yourself to go to the wrong school or worse, give up your scholarship and re apply if the school you are assigned to does not take you. I have this problem. I applied late and was lucky enough to get it but didn't get into the school I was assigned to. My first pick that I did get into was full. Don't force yourself to fight for a scholarship that you earned and to mess up your summer doing it. The medical stuff can also be a pain (see my thread), so be careful. Despite all of this, I still think that it's worth it if the military is what you want.
Good luck.