You have to send away by mail, so there is no way you'll get one before tomorrow. FWIW they are quick to turn it around.
One of my coworkers who wheels in this area a lot and attends many land use meetings suggests that if you want to attend a run, but do not have the permit back yet, it would serve you well to fill out the application form and then make a copy of it, the signed check, and a stamped envelope before you send it. Then carry it with you. Supposedly law enforcement will sometimes see this as a reasonable attempt and will let it fly, especially if the rest of the group is in compliance. This is what I was told by a credible source, but I have no experience.
Last time I did Box Canyon (2006) people generally were under the impression that as long as you stayed on the roads and didn't stop to picnic or camp you were okay. But reading some more recent threads by Tools R Us suggests this may not be true anymore.