I'm not sure that there is any vetting process. No government agency seems to be charged with vetting the candidates, and whenever anyone else tries to do it they are slapped down by the feds for a lack of "standing". The closest we've come to someone being vetted that I can recall is when Hillary Clinton pointed out that Songbird McCain could have been born in Panama (instead of in the US territory in Panama), which would have made him ineligible. Which is even weirder to recall because Ted Cruz's argument was that he was born in Canada to an American mother. But if Songbird McCain would have been ineligible due to being born in Panama to an American father, didn't Cruz out himself as ineligible?