Question: if the vehicle is off, would it still be subjected to the EMP?
Dumb question, just curious. Also doesn't an EMP blow out capacitors? I think I have a few of those in my Series 3...doing...things.
This is going to be grossly oversimplified, but the answer is "it depends, but probably not". Here's why. An EMP is just a moving electromagnetic field. When a conductor encounters a changing EM field, voltage potentials - and thereby current - can be induced. This is how generators work (and motors in the reverse). But this effect isn't magic - the orientation, shape, and material, and duration all matter a great deal.
In Russia, they had fires along power and phone lines from an EMP burst. Telegraph operators in the late 1800s had similar problems after a large solar flare. Long conductors separated by large distances can see large induced currents from these fields. So, for an EMP field of 30-50Kv/m and a large conductor, you can realize some tremendous amperage. Also, such blasts are pretty short lived - from nano to micro seconds. A nanosecond is about 11 inches long; that is that at the speed of light, you go about 11 inches in a nanosecond. A large conductor has time and distance to build up a lot of charge over a long distance.
Small electronics are subject to the same effects, and are more sensitive overall. An extra 2 amps won't hurt your telephone wire, but could melt a circuit board. However, because electronics are much smaller, they don't see the same currents as large ones do. The potentials just aren't there. Plus, they are more likely to be oriented in such a way that the currents generated by the field are negligible in the aggregate. So, the potential for ruination exists, but it is far from guaranteed.
Most car electronics aren't super hardened. But they are encased (usually) in some metal container and wrapped up inside the sheet metal of the hood/fenders, etc. That's a fair bit of shielding for the computers themselves. But, so too, the wires that feed them - they are wrapped in other wires (usually) and also contained within the chassis. That's not perfect shielding, but it should be good enough in most cases.
