Can you wear a sports mouthguard if you've got permanent dental crowns?

Yes. A sports mouthguard as well as other oral appliances(snore/sleep apnea, night guards, etc.), can be worn over crowns or caps. They can be made to fit normally and protect the teeth and gums. They also limit concussions. There could be a problem fitting an old appliance over a new crown because of changes in shape or size. .
Definitely, yes. Properly diagnosed, designed, and custom fabricated sports mouth-guards are essential in the prevention of athletic oral/facial injuries.and permanent dental restorations (e.g. crowns, veneers, fix bridges, etc.).
Yes, you can. A properly fabricated, custom sports mouthguard will protect your natural teeth and any prosthetic teeth.