Coordination can be trained contrary to popular belief; however, you probably won't find many coaches that are going to do much to help if the kid is lacking some fundamental, they feel strongly about.

Development timelines can also be relevant; kids don't get their crystalized reflexes until they stop growing and start building muscle.

This is one of the reasons tall kids often get a bad rap on coordination. Growing limbs just keep changing it's hard to develop tight muscle timings.

My uncoordinated boy does martial arts and will start running this fall. He has never done swim team but I have him swim laps for exercise. We have never pushed team sports for him because he never had any interest and it was clear he does not have a lot of natural ability there. My goal is finding activities that will help him develop better coordination and stay active.

It sounds like your son has found activities that he enjoys, I don't see why you need to add a team sport. Even at the rec level, by his age, most kids will have been playing for a while, so that could be challenging for an uncoordinated kid.

Have you considered martial arts? That might be good for someone like him and might help build focus and patience as well.

Cross country, track. Not because he’ll be good at them but because they are often good teams for kids of mixed ability.