If you buy a used car through a dealership then you need to buy auto insurance the same day you drive the used car off the lot.
If you buy a used car through just someone from their home then you also need auto insurance before you drive it and if you just want to haul the car back on a trailer or a car dolly then you can wait to get insurance on it when you get ready to register the car in your name.
You need insurance to drive the car and register it with a license plate and title it in your name.
So if you want to register and title the vehicle in your name you need auto insurance right away.