I have a case on disorderly conduct with a no contact order. There is a car in both names. If I made the payments can I pick up the car?

If the car is with the person who has the restraining order against you, the answer to your question is no. You would need to hire an attorney and your attorney can contact the victim for purposes of arranging to get your car, or if they are unlawfully holding the car from you, having nothiing to do with the current disorderly conduct or DV charge against you, you have the option if the car is valued under $10,000, of filing a small claims lawsuit for the return of the vehicle or a large claims lawsuit if over $10,000 for the return of the vehicle. To avoid the victim claiming you are contactinng them in any way, use a lawyer to help you.