Hi Tim!
Sounds like a fun trip! The only classic car rental company that have previous knowledge of is in Las Vegas. However I did get some search resullts.
I don't know where your start/end points are in VA or TN. But I was not able to find anything in Tennessee.
Hopefully one of these will be convenient for you.
Charlottesville, VA - center area of the state http://www.sportscarrentals.com/
Washington DC - borders VA http://www.capitaldreamcars.com/ This site was down when I tried to visit. Hopefully it will be running again soon.
Washington DC http://www.pure-exotics.com/
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