If you are not worried about the money get the chains,
@ArmyofMike is correct you must carrier chains on most mountain roads in California in the winter regardles of the vehicle.
There are four stages Caltrans/CHP use in snow conditions regarding who is allowed to drive.
Stage 1 - Winter conditions no restrictions
Stage 2 - 2wd with chains - 4wd with off road tires
Stage 3 - 2wd cannot travel - 4wd with chains and off road tires, or studded snow tires.
Stage 4 - closed to all traffic
It does not even require a stage one alert for all vehicles to be required to carry chains.