Could this possible make the Constitution Party as big as the Libertarian or Green Party is OTL? Right now, it's the smallest of the 3 big third parties by a lot, but this could give it more publicity and support among the far right.
Other than maybe a slightly bigger Constitution Party, I can't really see this having much of an effect. Maybe the Tea Party- if it ever starts ITTL- could coalesce around the Constitution Party with the Republicans as their second choice the same way libertarians generally support the Libertarian Party more than the Republican Party, leading to a slightly more moderate GOP, but even that's pretty unlikely.
The Reform Party probably still dies out. Unlike the Green, Constitution, or Libertarian parties, it didn't have a well-defined ideology that's attractive to people (it's candidates have ranged from Ross Perot to Ralph Nader), and it would never be able to accomplish its goal of fighting corruption or reforming the political system, since it can't get elected to national office (its only real success was Jesse Ventura's election) to do anything. The only third parties that can survive in the USA are the ones with an attractive ideology that isn't represented well by one of the two big parties (the Libertarians) or the ones for people who are too radical for one of the big parties (the Greens and Constitutionalists).