You can check with your DMV to see if they allow a search of a car you do not own. Most do not, unless you are a car dealer, but sometimes you will find a sympathetic agent at the counter.

However, if you have the description of the car and the license plate number and plate year, you may be able to get the VIN for the car if you can't get the entire title history. And with the VIN, you can start searching eBay and Google, and possibly obtain a title history.