How bad does wildfire smoke have to be to interfere with driving a normal car?

There can be many variables. Thick smoke that makes it difficult to see for any distance ahead would be the main one in my opinion. You need to be able to see if there is a vehicle in front of you & to be able to make sure you stay in your own lane. Seeing a fallen tree or limb could be very important as well.

With an internal combustion engine, outside air is needed to cause the combustion to take place.

If the outside air is too polluted (particulates, etc.) then the air will not be clean enough to have a good burn in the combustion chamber. And you will stall.

It is not the smoke but lack of oxygen, most wildfires don’t consume all the oxygen.

It’s not going to stop the engine by oxygen deprivation unless there’s a firestorm nearby that might just as well destroy the car with heat and flame.

Long before that, there’s an effect from low visibility, with thick enough smoke, that at least requires slower driving, and could get drivers lost and stuck in an area that’s burning down and having roads blocked. I think that’s the main thing just from smoke.

If enough soot or ash is in the air it can potentially clog up your air filter and hurt performance but it’s very unlikely to cause your vehicle to stall out.