Social security numbers are not being issued to H4, L2 visa holders. They can’t get SSN just for banking, driving, etc. Some states require that the person must have the social security in order to get the driver’s license. Social Security Administration(SSA) has given a final rule saying that SSN can’t be issued for getting driver’s license. SSN should be used only for tracking person’s earning over life time and driver’s license has nothing to do with SSN. If you are staying such state, you just can’t get driver’s license.
Very few states like Iowa may still provide social security numbers but you cannot use it for work purposes, kindly check with authorities.
Addition Information from Sands (thanks!) : You can write to SSA requesting for a denial letter and present it at DMV!