There is an NFL Coaches Association (like the NFLPA), but it is non-union. Coaches would be dis-incentivized from forming a union based on the tenuous way the coaching carousel works in the NFL. As head coaches are fired/quit/promoted, the opportunity for assistants to become head coaches means they wouldn't want to collectively bargain their rights.
Plus, I believe that Head Coaches are effectively "front office management", and the assistants are all employees of their respective team. A union in an environment where employees are fired/promoted up and down from management to regular employee almost at the drop of a hat would be unstable, and its not in the coaches' best interests to support one.
In the case of Sean Payton, that kinda sucks. He has no recourse for his suspension, because the league is the arbiter of that discipline. Well, the league is also the one who suspended him, so he's screwed.