I see credits for the music and lyrics here but not who actually sang the theme song.
shareIt was sung by an unidentified male chorus, probably employed by the production company. For more info, go here:
https://en.wikipedia.org/wiki/Maverick_(TV_series)#Theme_song
Any sufficiently advanced technology is indistinguishable from magic. - Arthur C. Clarke
I read somewhere that it was The Sons of the Pioneers, but I don't think it sounds like them. Keep meaning to play some of their videos on YouTube and see what I think then.
Some series did give on-screen credits to their theme vocalists -- e.g., the theme for Wyatt Earp was done by the Ken Darby Singers. I like it when they do that, but I guess it's a legal matter, and if the vocalists happen to be under contract to the studio, they probably don't get credit, as ZurichGnome conjectures.
OK, one mystery is solved! The Sons of the Pioneers did record the Maverick theme song, which you can listen to here:
https://www.youtube.com/watch?v=cSAGzBuf6Pc
BUT that's a considerably different arrangement from what was used on the show, done more in the Sons' usual style. So yes, they did "do" the Maverick theme song, but that's not what was used on the show.
As ZurichGnome says, the show's version was probably recorded by some anonymous contract group. After all, the Sons were very popular at that time -- why would the studio pay their (presumably much higher) rates and then not "brag" by giving them screen credit?