Never drive a car which is larger than a football field during a solar storm
Otherwise, don't worry about it.
The answer to Does a geomagnetic storm visibly deflect a compass?
shows the plot below which was a very large event.
It looks like the fastest change was about 200 nT per minute or about 3 x 10-9 Tesla/second. For a 6 square meter car that's an electric field around the car's perimeter of about 20 nanovolts all the way around, inducing a minuscule current, probably below that due to galvanic effects (dissimilar metals, rain, salt, rust, dirt).
You need a big antenna to pick it up. Since the area increases faster than the perimeter, if your car were 2,000 km by 3,000 km that would be about 20,000 volts! Of course a real circuit (a continental-sized power grid) presents substantial conductivity, so you wouldn't necessarily see 20 kV lightning, but you will see blown transformers and switching stations because of the induced current overloading.
Of course, if your car has a magnetic compass stuck to the windshield (my grandfather's car did) it might show several degrees of deflection, but that's probably buried in the errors caused by the soft-iron effects of the car's magnetic frame itself.