Although not all vehicle insurance providers will ask for your SSN as part of the pricing or application process, the great majority will, particularly if the insurance rates are credit-based.
Some insurance companies can insure you if you have a valid driver's license. Obtaining a driver's license without a social security number can be difficult since most states require drivers to have a Social Security number in order to obtain a driver's license. There are 16 states that allow people to apply for a driver’s license without SSN, which includes California, Colorado, Connecticut, Delaware, Hawaii, Illinois, Maine, Maryland, Nevada, New Hampshire, New Mexico, Oregon, Utah, Vermont, Washington and Washington, D.C. You can contact your state's DMV for more information.
The information in this answer is general in nature and is not legal or insurance advice for your specific situation. Policy language varies carrier by carrier; your declarations page is the final word. If you're unsure, call your agent before you need to file.