My son works about 15 hours a week at a private country club and he has not received a paycheck in 4 weeks. Do they have a right to delay his pay?

No. But paydays need to be defined for the job. Its possible he is scheduled to be monthly.

Yes, he needs to ask when are paydays, when are pay periods, if he started for example on the first day of a new pay period, and they are paid every two weeks, he would not get a pay check the first pay date, since that check would be paid the next. So going 3 to 4 weeks for pay is very common.

He just needs to ask them.