Chan and cars are not usually associated with one another. He did pick up a driver (Birmingham) in the later Monogram films.
However, there is a famous scene in Black Camel where Chan is shown driving (badly) with Bela Lugosi as a passenger. Bela is obviously nervous especially as Chan is commenting that he and road now turn at the same time! Great stuff.
Charlie Chan drove a 1919 Ford Model T until the final scene of "Charlie Chan's Greatest Case" where he and his "multitudinous family" drive off in their new 1933 Ford Touring Car, a gift from a very appreciative Minerva Winterslip. As Steve mentioned, we see the Model T in a couple of scenes from "The Black Camel." There were also a couple of humorous scenes in "Greatest Case" where Mr. Chan places a locking device - a forerunner of "The Club"? - on the rear wheel of this car, to protect it from theft!
If any one car is connected with Charlie Chan over the length of the series, it would probably be the 1919 Ford Model T...