So, you're not really roughing it, and you're on the beach for $15/night (April 1997 prices.)
To make reservations, call the above number and reserve your spots. To hold them, you need to get them a one night's deposit. You can mail it either to Mexico (expect the mail to take at least 2 weeks):
Playa Elegante
ADPO Postal 101
Puerto Penasco, Sonora
Mexico 83550
Or, if you're really in a hurry to get it there, you can mail it to their US address in Lukeville:
Playa Elegante
P.O. Box 56
Lukeville, AZ
85341