In fact, in the US, the MLS (pro soccer league) is still playing games during the World Cup, even though some of their players are missing. I believe about half the players on the US team come from MLS (the rest are good enough to play in other (European) leagues).
The MLS, unlike most of the world’s leagues, runs during the summer at the moment. This is largely due to the lack of dedicated stadiums. In some cities, they’re using the (American) football stadiums for their games (the NFL runs from September-January). There are plans (and pressure from FIFA ) to change this, with dedicated stadiums or new leasing agreements in the next few years. Most likely, the MLS won’t be playing during the next World Cup (whether or not the USA qualifies).