Get a camcorder. Webcams have very little dynamic range and just can't handle the difference in brightness levels you'll see at the track.
I use Viosport analog cameras and solid state recorders for exterior cams and I use a Sony HC3 camera for in car with a wide angle lens on it. This camera is small and does a great job handling the brightness range between inside and outside the car.
If you want to see what the video from the HC3 looks like you can check out this video we recently did at VIR (Its not exciting, just some recon laps ahead of putting students out on track) - http://www.youtube.com/watch?v=OOFbExqa … annel_page
To get the best picture you will want a shock mount to help dampen out vibration. Most cameras you buy now will have a CMOS imager and vibration will seriously screw up your image due to "rolling shutter" that is part of the CMOS design. I would recommend against the small Canon camcorders for this very reason. Based on my personal experience using the Canon HF100 these cameras are unusable in a race car. The Sony cameras I've used seem to display LESS of this effect. If you can afford a camcorder with an OIS (optical imager stabilization) you will get better images than a camera without. Not all stabilizer systems are created equal. Some handle the high frequency vibration (race cars) better than others. My Sony V1U does a GREAT job with vibration but its pretty expensive to be putting in harms way. Its not small either.
You will want to make sure your setup has a safety strap that will prevent it from hitting the driver in case the mount fails during an unusual event. I will assume this will be required to pass tech.