My 360 controller works natively, without a driver. Are you running XP? I'm on Vista, that may be the difference.
The 360 controller works great for this game. One minor tweak is needed to get you going: go into the filter settings for the steering axis in the controls panel, and increase the curvature quite a bit. Otherwise the car will self-steer and be way too twitchy.
I recommend binding downshift to A, and upshift to X. I'm not talented enough yet to work the clutch too, so I leave that one on auto.
The scary thing is, I realize there's nothing keeping you from driving a real car with a game controller, if you went through the trouble of robotizing the steering, brake and throttle. Why not? Maybe someday we'll look back on driving cars with your feet as old-fashioned.