In The Legend of Korra (second Avatar Series from Nickelodeon) professional tournament bending has a trio of earth, water, and fire benders squaring off against one another; but why isn't airbending represented?
Avatar Aang is the last of the Air Nomads, a group wiped out by the Fire Nation. He had three children who could have been airbenders:
Kya is the first of Aang and Katara's three children and their only daughter, as well as the only waterbender that the couple parented.
Kya and most likely her children are unable to airbend, so can be discounted.
Bumi is Aang and Katara's second child and oldest son, as well as the only non-bender among the couple's three children.
Bumi is a non-bender, so he and his lineage can likely be discounted.
Tenzin, the youngest son of Avatar Aang and Katara, is the only airbending master, and represents the Air Nation on the United Republic Council. He resides on Air Temple Island,... with his pregnant wife, Pema, and his three children, Jinora, Ikki and Meelo.
Tenzin is a master airbender and until the first few episodes looked down upon the Professional Bending tournament. His three children, though all airbenders, are not skilled enough to be considered for a team (And likely too young - though who knows the rules, if Toph could previously compete in a similar tournament?)
Tenzin lives on the island with Air acolytes, but none of them are airbenders:
Air Acolytes are monks and nuns that reside on Air Temple Island. Although not airbenders themselves, they carry on the teachings of the Air Nomads, their culture, and their traditions through practice.
So as far as we know the only air-benders in existence are Tenzin and his children. None of who are eligible or willing to join the tournament.
Bear in mind that there could be a team with an airbender that we haven't seen, but it's highly unlikely.
Further to this, given that in the current rules there is one (and only one) representative for each element required for a team. If an air bender was needed, there would need to be at least as many air-benders as teams, which is unlikely given Tenzin and his family's status as the last air-benders.
Quotes taken from the Avatar Wikia
It's likely that when Pro-bending was invented that the only airbenders were Aang and Tenzin. If Air bending was to be represented all pro bending teams would have to have an air bender. There are quite simply not enough air benders yet for it to be represented in the sport. I think it's safe to say that the only air benders during the popularity of pro-bending so far are Tenzin and his offspring. In "Welcome to Republic City" Korra says she would, "love to find another air bending master but [Tenzin is] the only one." It could be that if Kya and Bumi have children, they could also be air-benders but if they were, they are probably too young to know that they have the ability. If they are older it's unlikely that they are airbenders because they would be training with Tenzin if they were. The claim that Tenzin and his family are the only airbenders in existence is supported in "Turning the Tides" when Lin Bei Fong says to Tenzin, "You and your family are the last airbenders" Also in "Endgame" when Amon is about to take away Tenzin, Dinora, Ikki and Meelo's bending he refers to it as "rid[ing] the world of airbending, forever." further supporting that they really are the last airbenders. Counting only Tenzin and his children that's only four known airbenders. Which is not even enough to supply the 17 notable pro-bending teams not to mention unknown teams. Also pro-bending teams come from all over world, across the united republic of nations and the only place airbenders live are in Republic City's air temple island making it unlikely for more people to compete if they need an airbender to play. Other reasons might be because airbending might be a more difficult to include because a lack of understanding of the practice, the spiritual practice of airbending might also have principles that contradict competitive fighting. It might also be that air bending is difficult to put rules and regulations as the other elements are. Another element that is not represented in Pro-bending (probably because of ethical reasons for the most part and also that it's difficult to regulate) is Energybending.
Ang was the last air-bender alive in his time of the first Avatar Series.
Wikipedia's Last Air-bender page reports:
Fire Lord Sozin then carried out a genocide of the Air Nomads; the entire people were wiped out, leaving Aang as the eponymous last Airbender.
As a result, his family are likely the only airbenders left alive. Unlike the other groups, unless other air-benders were found, there would be very few air-benders in proportion to the other bending groups. This would probably preclude there being enough air-benders around to form a league or to be members of the Professional Bending Tournaments.
(This review contains SPOILERS for like, the first major point after Season 2. MAJOR SPOILERS SERIOUSLY MAN IF YOU HAVEN'T WATCHED SEASON 3 GO WATCH IT)
Like everyone else says above there were only five living Airbenders not including the Avatar. 4 of them are too young to participate in Pro Bending (Tenzin's kids), and Tenzin wouldn't have joined for sure.
(here's the spoiler-->) However, with the recent development of all those nonbenders who received Airbending, I don't see how, once they've been trained and stuff, Pro-Bending can't be altered. After all they wouldn't even have to do much to create another spot on the field for them because there's always air everywhere. The only problem is that if they did that, all teams would then be required to get an Airbender, and there aren't enough of them yet to supply the spots, who would want to be in Pro Bending. It'll probably be another generation or so before anything like this happens, since Airbending isn't widespread enough yet. This entire paragraph is speculation.