Does having a sunken chest stop a child from doing any particular sports?

No: Not anymore. My brother had a steel plate put in his chest as a kid and wasn't allowed to play sports for over a year. Nowadays, nothing is done. It doesn't affect the lungs and there are no restrictions.

No: Unless the child has specific symptoms it should not.

Depends: You are talking about a pectus excavatum. Normally these children have very few in the way of symptoms. They can do any sport contact or not. If exercise intolerance becomes evident then they can be evaluated with puilmonary function tests to indicate if surgery to repair pectus would help.