You cannot build a modern car engine without chips, let alone a car. Also, no way to hit safety requirements without chips (air bag deployment for example).
I mean yes, if you take that tautologically, since a modern car engine has a chip. But probably you could build an engine that meets modern requirements without. People forget how clever analog solutions can be. It would be a huge pain in the ass though, and take years.
No, not since the late 80s when electronic fuel injection became necessary to meet emission standards. The alternative, carbs, just isn't good enough anymore.