How are these competition organized? Do both amateurs and professionals play there? If professional players play also for league teams, are the league matches scheduled in such way thad they do not to clash with ATP, WTA and ITF events?
In Austria there are amateurs and professionals playing. Making the league matches not clash with ITF events is virtually not possible because ITF tournaments happen every week.
I would think that they schedule the matches in a way that they don't clash with major tournaments (ATP 1000 and grand slams). Maybe they also prevent clashes with ATP 250 tournaments which are geographically nearby. In some cases, the league is also played in a short amount of time, so players don't have to spend too much time for the league and to make scheduling easier.
I am also pretty sure that geographically close leagues somehow coordinate their schedules. Because players possibly play in several leagues (for instance in Switzerland, Austria, Italy, Germany, ...) they should be able to do that without conflicting league schedules.