What do you do when you are involved in an accident that was not your fault but your insurance has lapsed and the other driver wants your information?

Tell that driver exactly that. And see what type of arrangement, you can come up with. In most states, your license will be suspended for a year, so if its not your fault, explain to the driver, even with your information it will not tilt the scales in his favor with the insurance company, but surely damage your license.