There is no reason to believe that Norman was a member of Starfleet, but instead just someone who snuck on the ship wearing a uniform. Norman was impersonating being a member of Starfleet.
Well, I think there would have to be a little more than that, he would at least have faked transfer orders. McCoy even mentioned that there was something not right about him before he took over the ship. I don't think he just put a TOS uniform on and popped in without question.
Here is an amusing dialogue between Spock and McCoy talking about Norman. It's funny to see that McCoy's instincts and not Spock's logic were correct in this situation.
(Spock and McCoy are greeted by a crewman in blue as he walks along.)
NORMAN: Good morning, sir, Doctor.
SPOCK: Something wrong?
MCCOY: Yes. There's something odd about that man, and I can't quite pinpoint it.
SPOCK: Perhaps you're making a rather hasty judgment. Mister Norman has only been aboard seventy two hours.
MCCOY: I know when something doesn't strike me right, and he doesn't.
SPOCK: Specifics, Doctor. Labels do not make arguments.
MCCOY: All right. There's something wrong about a man who never smiles, whose conversation never varies from the routine of the job, and who won't talk about his background.
SPOCK: I see.
MCCOY: Spock, I mean that it's odd for a non-Vulcan. The ears make all the difference.
SPOCK: I find your argument strewn with gaping defects in logic.
MCCOY: Maybe, but you can't evaluate a man by logic alone. Besides, he has avoided two appointments that I've made for his physical exam without reason.
SPOCK: That's not at all surprising, Doctor. He's probably terrified of your beads and rattles.