I have never done it but here is some info from wdwinfo.com:
Experience the thrill of being a passenger in a two seater stock car while the professional driver reaches top speeds of 145 mph. Offered on a first come, first served basis, you must be 16 years old with with parental consent required for guests ages 16 - 17. Available every day from 9:00am to 4:00pm.
$116.09 - thru December 31
DH just did it in June and loved it. The shuttle picked us up at the TTC. Then we signed the waivers and stuff. I believe it was about $120. They sent DH to put on the driving suit. We walked outside and they asked him more questions. The put the helmet on him and such. Helped him in the car while the driver was already in there. A photographer took his picture and off they went. I held my breath the entire 3 laps. They let us take as many pictures as we wanted. I believe they do only ride alongs on Tuesdays and Thursdays.