No, insurance depends on your previous fuckups, so you definitely need DOB and SSN. Otherwise they might misquote you, which can end up in court.
A "soft" quote based on general information.
A "hard" quote based on very specific information.
They'd never convince me they cannot do this, especially with all the extremely misleading commercials on television.
A soft quote would be just fine for me and probably many others.
A smart thing to do here might be to figure out a way to do progressive quotes/queries; take the DOB+zip+car information just to start, then let the user decide whether the variance in the results (which will be the best-case ones) merit coughing up more info.