Good question.
I would say there's a lot of groups that are anti-communist but even some of those who used to be pretty stringent are now pushing various forms of populism, which is essentially Communism-lite.
For example, Heritage recently changed positions and now say that a free market is not the ultimate goal, but a civil and secure society. So if the market isn't facilitating such a society, government should make it so.
Even the group I used to work for has started off in an odd direction. It used to be fairly libertarian then pushed a warhawk (among other problems) for president.
Crazy that I have to sit here and think this hard about who, as a groups, may be the best positioned to really push back.