Is there a safe way to operate any type of a tesla coil on a functioning modern car (moving road vehicle)?
Unlike the commentators, I don't think there's any inherent reason why this wouldn't work, provided you don't overdo it.
The obvious risks are:

arcing
conducted EMI into the car's electrical systems
radiated EMI into the car and nearby cars
electrocution risk to operator
fire risk from ignition of fuel

I'm assuming the coil is outside the vehicle (on the roof? In a truck bed?) and of small to moderate size and fixed appropriately. The coil's ground should be properly connected to the bodywork of the car or at the very least to the car's electrical ground.
Strong electric fields move to the outside of metal objects. So fields relating to the coil will be kept outside the bodywork of the car. Petrol cars are already very electrically noisy devices with multiple-kilovolt sparks of their own, so provided you don't inject anything onto the positive side of the car's electrical system the existing shielding should be adequate.
Having a large enough coil to arc to other people's vehicles should be avoided - not necessarily because of electrical risks but because this will be extremely distracting. You may also run into police restrictions on &quot;lights&quot; on the car. And watch out for bridges, overhead wires, and other street furniture.
Oh, and make very sure the thing is off and discharged through a bleed resistor before you go anywhere near a petrol station. I think that might actually be the biggest risk.

some kind of electrical cage around the tesla coil?

A grounded mesh cage for the coil to arc to would be a great improvement in safety.