Unfortunately, in the state of Missouri, insurance is tied to the vehicle, not the driver. State law requires that all cars are both registered and insured, and it is illegal to drive a vehicle without insurance. There are a few non-owner insurance provisions in Missouri law, but these are limited to rental insurance policies and non-owner insurance policies for minors in foster care homes.
It is very important that you resolve your current insurance situation as soon as possible, because if you are stopped by police and cannot show that the car is insured, you and your children's father may face penalties. The car may be impounded, you may have your license suspended, and he may face a fine.
The best solution in your case would be to convince your children's father to purchase insurance for the car and list your name in the policy as a covered driver. Alternatively, you could speak with him about transferring ownership of the vehicle to you so that you can purchase a policy for the car in your name.
At minimum, according to Missouri state law, you need to have liability coverage and uninsured motorist coverage for the vehicle. Limited liability insurance will cover any injuries or property damage caused to others by your vehicle. Uninsured motorist coverage is intended to protect you in the case of an accident with an uninsured driver.
If none of these options work in your situation, the last possibility is to complete "Proof of Financial Responsibility" declaration, which is a legal declaration that you have the necessary funds to cover all medical and property damage expenses should you be involved in an accident. To do this, you need to submit a surety bond, real estate bond, or cash deposit of 60,000 dollars to the Missouri Department of Revenue, and they will issue you a certificate of self-coverage. This is considered an acceptable substitute to car insurance in the state of Missouri.