Spock was not the first full blooded alien in Starfleet. He was half human.
Captain Garth of Izar who was kirk's idol when Kirk as at the academy looks human but apparently is native to the star Izar. Since Garth looks so human, it is possible that other human looking starfleet members are also aliens.
KIRK: I'm sorry, Lord Garth.
GARTH: You Earth people are a stiff necked lot, aren't you?
And the captain and/or other senior officers of the USS Intrepid probably joined starfleet before Spock.
UHURA: Captain. There was a message from Starbase Six. Heavy interference. All I get is Intrepid and what sounded like a sector co-ordinate.
SPOCK: Interesting. No indications of magnetic storms in this sector.
KIRK: Try another channel, Lieutenant.
UHURA: Aye, sir.
MCCOY: The Intrepid is manned by Vulcans, isn't it?
Commander Sonak in Star Trek: The Motion Picture may have joined Starfleet about the same time as Spock or possibly earlier.
In the Alternate universe of "Mirror, Mirror" Three may be other Vulcans on the Enterprise:
SPOCK: If I am successful, you see yourself a step nearer to the captaincy. I do not want to command the Enterprise, but if it should befall me, I suggest you remember that my operatives would avenge my death and some of them are Vulcans.
And I think I saw a Vulcan with Spock. Some of them may have joined starfleet in the main universe.