State notification systems differ for contacting authorities if you are aware of someone driving without insurance. Most all states require auto insurance to be placed on a motor vehicle or another type of financial responsibility be secured on the automobile so they do accept information from parties who know of a person driving illegally without insurance.
In some states there are Financial Responsibility Units or Departments which you can call and report someone that you suspect or know is driving without insurance. In other states you report this type of driver or car owner directly to the state's Department of Motor Vehicles.
For example in New Jersey the state encourages NJ residents concerned about insurance cheating and who have information about insurance fraud to contact the Division of Criminal Justice - Office of Insurance Fraud Prosecutor toll-free hotline at 1-877-55-FRAUD.
In Florida you would contact the FL Bureau of Financial Responsibility. This state agency was started to ensure that people are responsible for their actions involving motor vehicles. This agency does so by making sure everyone has insurance to pay for the damages they cause, or they may cause, when they own a motor vehicle or register a motor vehicle.
So it varies in each State how you would go about reporting an individual that you are aware of that is driving a vehicle without insurance. In most States though if you contact your local Department of Motor Vehicles you will be directed you to the state agency to report the uninsured driver if the DMV if not the correct bureau.
No matter what State you are located in, if you know of someone whose vehicle is uninsured and it appears that the DMV has not yet been informed or taken action against them, you can try to report that vehicle owner to the state. Typically the DMV has a consumer hot-line that you can call into to ask questions or report problems. They might be able to take down the information you have or forward you onto the appropriate state agency.