Do you have any legal recourse if you were hit by an unlicensed and uninsured driver?

You have recourse but it may not immediately result in monetary compensation. You can certainly try to get the at-fault party to voluntarily pay the damages. You can also file suit and if the court finds in your favor, obtain a judgment. Most states have "financial responsibility" laws requiring auto owners and drivers to maintain insurance. If they don't, the state can suspend the driving and registration privileges of the person until the judgment is paid. Since, on these facts, the person did not have a license, there would be no license to suspend, but if he/she later tried to get one, or to register a vehicle, payment arrangements would have to be made with you before a license or registration is issued. Keep in mind, though that a judgment has a statutory "life" (for example, 7 years) that varies by jurisdiction. If the life of the judgment expires without being renewed, it loses force and effect.