I have worked for four car dealers in my life that ran credit. A Subaru-Kia-Isuzu dealer, an AutoNation used-only store, a Honda dealer, and a Saturn dealer.
None of the four could run credit without an SSN. Now this was in 1997, 1997, 2003, and 2005. You could swipe the DL to see if it was VALID. You could enter the name and address in the computer for mailing list purposes. But you couldn't get the beacons or anything until they signed a credit app.
I tell you what, I really enjoyed working at the AutoNation store. The Honda store was awful and the Saturn dealer was worse, if you'll believe that. It was NOT representative of other Saturn dealers, though. The import store I only got to work at for 3 weeks, sold 7 vehicles, though I was doing ok. But then it got bought out by a large dealer group and anyone without 6 months tenure was given their final commission check and told to hit the road.
It may be possible for a dealer to pull credit without an SSN. But it would be stupid, illegal, and immoral. That said, they are car dealers after all. And they're why I scour CL and impound auctions these days. I haven't bought a vehicle off a dealer lot since 2006.