I can't seem to find this movie in the database. Here is the imdb listing:

http://www.imdb.com/title/tt0039066/

It's not exactly packed full of cars, in fact I only saw two. They were both convertibles and I don't know enough about 1940's American sheet metal to even venture a guess on make and model. The first convertible that promopted me to look up its listing had very unusual door handles, they looked almost like a round push button switch.

It is playing on TCM and sometimes they show some pretty obscure movies not readily found on DVD or even home video, so it may be hard to find.