Yes, Marlowe is an attorney, and he can tell you that even today, many court documents that need to be filled in and filed with the court come on a pad. An attorney can buy a whole pad, but he has to be on the ball, because they are often updated, and then the pad he bought, is no longer accepted by the court.
Anyway, for birth certificates back then, it is possible they just came on pads from a printer and were torn off when needed, filled in and filed.