I'm sure that, if you got strictly technical about it, the people who've "played" the most characters would be the uncredited stunt doubles and stand-ins, or the voice actors they brought in to do the comm voices of unseen characters. (While rewatching Enterprise
as research for my latest Trek novel, I noticed that all the comm voices in a certain stretch of episodes all sounded like the same guy.)
By the same token, if we include the animated series, then James Doohan, George Takei, Majel Barrett, and Nichelle Nichols have dozens of characters each -- Doohan the most, though unfortunately the '90s edition of The Star Trek Concordance
incorrectly credited him with many characters played by different uncredited actors (the main one of whom may have been Lennie Weinrib, who did most of the supporting voices in Filmation's '70s Batman
series) and sources like Memory Alpha have copied that mistake.