Yes it's possible. I am running currently amateur rapid chess league season 1. It's kinda like what you are saying. It's rapid 15+2. A player gets about a week to play a round. He has to negotiate with his paired opponent and decide a time and then play the round at that time.
It's a team event. 4 players a team. Team rating shouldn't be more than 6600.
Hi. Is it possible to run a (club) tournament on chess.com with "live" time controls (e.g. blitz or rapid), but where the rounds can be delayed so only one round is played per week, for example, or even where the players of a particular game organize a suitable time to play?