"So I basically lied to him and said 'Oh yeah, it's Andorians and Tellarites.' "
Michael Sussman was instrumental in establishing which species founded the Federation. "My phone rings and Brannon [Braga] says, 'Mike, real quick - who are the founding species of the Federation?' What flipped through my mind," recalled Sussman, "was that it had never been established, but there had been a lot of fan speculation going back to 'Journey to Babel' that many of those races - the Tellarites, the Andorians and the little gold guys - were among the founding members. For that quarter of a second I'm thinking, 'What do I tell him? If I say it's never been decided he might make up a couple of new races and that might set certain people off'. So I basically lied to him and said 'Oh yeah, it's Andorians and Tellarites.' He put it in the script and now it's canon!" (Star Trek: The Official Starships Collection, issue 37, p. 15)
And then you basically see neither of them again after TOS and some films until Enterprise. They definitely aren't in TNG or Voyager and if they're in DS9, it's in the background where I never noticed.
There are only two Andorians shown in TNG: one was a holographic depiction as a possible appearance for Lal, and the other was in the background on Risa.
AAAMAZZARITES – Therbians from planet Aaamazzara. They generate their own clothing from out of own mouths, like bees making hives. They manufacture everything they use from their own chemistry, from inside their own body, from clothing to furniture.