Do insurance companies cover an accident when it is your fault and you don't get the other drivers information for the claim?

Yes, you would be covered by YOUR insurance company, given you have "Full-Coverage" coverage on your policy. Keep in mind, when filing a claim with your own insurance company, that this claim will follow you on a permanent insurance record for 5 years, possibly more in depending on your state. So be very wary of claiming an accident that was someone else's fault on your insurance. This may raise your premiums, or make you ineligible for insurance through A rated companies. If you have any more questions, direct them to my email. Sean