Does being in the military mean his vault certificate is public record somewhere?
Probably not, since the military has the same sort of "Privacy regulations". One woudl need access as part of one's duties to be able to see it.
Ironically, it was probably a "public record" when it was generated. But so was Obama's. The rules regarding "privacy" are more recent than that. It's not really a privacy concern at all, but rather one of preventing identity theft.
Which is why I don't think the rules should apply to major public figures, since anyone applying for a driver's license, or other ID document, would get a pretty good examination if they claimed to be "Barack Hussien Obama", CLB/BC or not. Be a little different with "John McCain", but with "John Sidney McCain III", maybe not much. Of course most folks JSM's age are not looking to get hired into a new job, they are either retired or looking to retire, and if they tried to claim his Senate retirement or his military retirement, they'd again get the same sort of proctological exam that someone presenting themselves as BHO would.