For a prevailing wage job?
They don't need to be registered, but it would be very expensive not to be. Only registered apprentices can be paid as apprentices on a PW job, meaning if they aren't registered they must be paid full journeyman wages. So, if you send a green first year apprentice to the job who is unregistered, you have to pay them full rate. In order for them to be registered they need to be actively enrolled in a state approved program.