According to Chip Foose's website, you can find the Overhaulin' nomination form at Overhaulin.tv. It appears that you cannot apply for yourself but need to be nominated by family or a friend. Overhaulin' seems to place an emphasis on returning veterans and those hardest hit by the economic downturn.
A central premise of the show is that the owner of the car (usually referred to as "the mark" or "the victim") does not know what is going on until the car is returned and all is revealed. Foose and his team go to elaborate lengths to acquire the vehicle with the help of the owner's nominating family and/or friends (referred to as "the insiders"). Sometimes, the car is "stolen," sometimes impounded by the show's team posing as police or repo men, and sometimes it is taken into a shop for some routine repair and "misplaced." The car is then overhauled by Chip Foose and his "A-Team" of mechanics over the course of a week before being returned to the owner for a dramatic reunion between man and machine.
Overhaulin' was originally a TLC production, running for five seasons from 2004-2008 before being cancelled. It was revived by Discovery subsidiary Velocity in 2012, where it has so far run an additional three seasons.