Should I allow my sixteen year old son to wear contacts?

Depends on him. Is your son responsible and able to follow the directions for their use? If so, he may be fine wearing them. Children as young as infants have been able to wear contacts. The key is removing them and cleaning them appropriately. Abusing contact lenses by not cleaning them or overwearing them can lead to intolerance and even infections. Be a good judge of his abilities and ask his eye doctor.
Know your child. It really depends on the maturity level of the child. Improper lens care, poor hygiene, and overwear can lead to permanent vision loss. I would only allow under very close supervision and would not allow extended wear (sleeping in lenses).