They did take part in the qualifiers last year with a team named Tbilisi Caucasians, a select side with the best players from all the Georgian domestic clubs. They were beaten both home and away by Italian club Rovigo and failed to qualify. I guess their union decided it wasn't worth the cost to try again this year, due to Georgia's location travel costs a bit extra.

Also, World Rugby doesn't fund club competitions. Clubs that play in these qualifiers do it from either their own funds or are supported by their unions.