This summer, Audi held an elaborate media event in Spain to celebrate a new sedan that would allow drivers to let go of the wheel and pedal while in traffic jams and pay attention to something else.

The driverless technology heralded a new era for mainstream auto makers in their pursuit to create fully autonomous vehicles. But the rollout by Volkswagen AG’s luxury unit—with production for the U.S. originally planned to begin this year—is now uncertain, as the car maker continues to wait for Congress to pass legislation paving...