There are many companies that do this, and there are open and closed carriers (closed being he most expensive).
This is one of the functions of my job, and the main companies I've used are Reliable Carriers, Intercity, Pilot Transport, Select 1 and Waggoners. There's also Horseless Carriage, but I haven't used them. Then there's BMT (Bavarian Motor Transport) but they only handle the east-coast moves. Sierra Mountain handles west coast moves.
Prices can range from $500 to $2500 depending on distance. Cross country is generally over $2K. There's usually a surcharge on top of the rate. Ask for a non-rush rate, which could take two weeks but will be the lowest price.
I think all have web sites. Call and see if you can get an email address for someone to bid on private retail moves (my contacts are corporate account managers). Then email all of them and ask for bids. Provide the type of car, pick up city, drop city, and approximate timeframe. They should all reply with their best bids. Good luck.