We did not find there to be a grace period for insurance coverage in which to drive home a vehicle after buying it from a private seller in Illinois. The information we found stated that before driving your "new" car home from the seller or the dealer, you must have insurance coverage.
To arrange coverage if you are buying a car from a private seller should be as simple as calling your insurance company and providing them with the required information on yourself and the car. Most insurance companies can fax you a temporary insurance card that you must keep in your car along with the registration card.
The only way there may be a grace period is if you are transferring coverage and plates from one of your other vehicles to this one. If this is the case you may be given time to switch the coverage off of one car and onto the car you are buying from private seller. This grace period for the transfer of coverage will depend on state laws and your insurance company's guidelines.
If you would like to find out more information on this topic you can contact the Office of the Secretary of State, Mandatory Insurance Division at (217) 524-4946.