Team sports are really hard for ADHD kids. They just don't have the speed the kids need to keep there attention. Soccer might be the closest but even that will be hard since he will have to sit on the sidelines for about half the game so everyone will get the chance to play.
I would like to suggest you try BMX.. It is Bicycle MotoCross. Racing a bicycle, that you peddle with your own feet.
There are tracks all over the USA and Canada that you would have access to. You can drive to a lot of tracks and race then drive to another one and race again.
All you need is a bike and a helmet. Pads are good if you need them or want to use them. We also have goggles for our girls helmet so she won't get dirt in her eyes.
There are tracks in UT, some are further away but some are close enough to drive on a weekend.
We have 3 tracks in the OKC area, a couple in Tulsa and further east in Miami Oklahoma, then there is one up in Ponca City that is not open right now. Then there are multiple ones in Texas and Kansas. We don't have a good vehicle right now but still like to do this on the weekends here in our area.
The reason I suggest this sport is because it is simple, you ride your bike, you compete against yourself, if you practice you get faster, there is no strategy, either you ride faster or you don't come in first. If you practice and keep trying you can win lots and lots of trophy's, ribbons, and even collect stamps to buy new gear and dress up fancier.
Our BMX gear is all hand me downs that are just about worn out now. The leather insets on the pants has been chewed up by the chain and is just about holey....
List of tracks in Utah.
http://ababmx.com/site/bmx_tracks/by_state?section_id=12&...
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The one is South Jordan isn't horribly far, close enough to not be out of reach by far enough away to think about spending the night and making a weekend out of it. There are 2 listed for this track, one is indoor and one out, but this may be just an oversight and they listed it twice.
http://ababmx.com/site/bmx_tracks/29?section_id=12
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There is a track in Richfield which would be too far for me to go but it might be something your family wouldn't mind doing for a treat. It seems this one is an indoor track so that is pretty awesome.
http://ababmx.com/site/bmx_tracks/538?section_id=12
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There is one on Tooele, it is west of the South Jordan one and a bit further but still, making a family treat out of it might be a wonderful thing.
http://ababmx.com/site/bmx_tracks/288?section_id=12
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Farmington appears to be on the other side of SLC.
http://ababmx.com/site/bmx_tracks/540?section_id=12
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And finally there is Virgin UT. It appears to be right by the Farmington one.
http://ababmx.com/site/bmx_tracks/540?section_id=12
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It is too bad there is not one right in your neighborhood but they are not across country too far. I would research it and see if you think there is a chance he'd like it. Our boy was going to start riding this year but we just can't travel in this vehicle very far. It's not safe.
The youngest I have seen a young guy ride was 3 years old, the oldest was in his 50's. But they do have riders older than that, women riders too. Our track had a girls race team, there were girls and women on the team and they won all kinds of awards.
I know it's not right in your neighborhood but please consider giving it a look. Driving up to Sandy, well the South Jordan one, and watching a race day would be a nice day trip.
Our track has the policy that a prospective racer can ride free his first race to see if they like it. They usually will have kids who race at different time and he can borrow a bike and the track almost always has extra borrower helmets. Give it a try to see, it is very simple, you wait until the gate goes down and ride, ride the track and go as fast as you can without sitting down at all, as you get better you can do the jumps and eventually learn tricks. This is an Olympic sport now.
The difference between a regular kids bike and aBMX one is the weight and there are no decorations on a BMX bike. Walmart does not sell real BMX bikes. They sell bikes that cannot be ridden on an official track. There cannot be a kickstand, no reflectors, no dangling items or mirrors, no horns, nothing that could come off and hit the rider. They can be bought but to start out see about borrowing one from someone who is racing and has some time between your race and theirs.
A lot of handy guys will just buy the frame then put tires, chains, all the part on themselves to make it just the way they want it.