Can you go to an amusement park with a broken wrist?
Just call and ask would be the best advice. Since it is his wrist, I do not seeing it being too much of a problem! I see people ALL the time getting on roller coasters with cast on their wrists... heck even I have done it.
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Most rides are not recommended for guests using casts for the safety of the rider. Riders must review each ride to consider the risks, and follow all posted warnings and operator instructions for the safe use of the ride.
In my experience, most rides can accommodate someone with their arm in a cast, but you will want to check with your doctor prior to your visit to see if there are certain things you should avoid. Water rides, spinning attractions, thrill rides, etc.
Where I work the rule goes like this, if the cast covers one joint it is okay, the rider still has flexibility to move their arm, leg, or whatever. But if it covers two joints, such as wrist and elbow, then the person can't ride. But it is always important to know your limitations, so you don't injure yourself worse.
Riders with one missing arm or hand may ride, provided rider has ability to hold on with one functioning hand and brace themselves with two functioning legs. Riders must have control of their leg(s). Riders with arm or leg casts shall not ride.
Take care when entering and exiting rides.Even with a broken arm, you or your family member should be able to go on most rides at Disneyland. Take care when entering and exiting these rides, and make sure to keep the broken arm inside the ride at all times, so it doesn't get bumped or jostled.