How'd they put it out?

The snow from the edge of the road...

I've always assumed they just waited and let the fire die out on its own, as it seemed to have burned just about everything that was flammable in the car's interior. Perhaps there was a fire extinguisher in the car's trunk (perhaps it's required by law for car rental companies to provide one in each car?), but in any case, neither Neal nor Del was in a state of mind to consider this.

This scene also would not have worked so well if it were done today, as any passerby, even at the late hour when there was sparse traffic, would have used their cell phone to call 911 and report the car fire.