Granted it's not a sick chinchilla, but we drive more then 2 hours to chin shows with them all the time. Keep the car cool, keep him in a small dark carrier, and he should be able to handle the drive. If it's a bumpy ride--take it slow and easy on the way home as to not cause him anymore discomfort then he is already having.
It may be worth trying, especially if this is the only option you have other then putting him down.
It may be worth trying, especially if this is the only option you have other then putting him down.