The best answer to this question is NO...
Every vehicle is designed both internally as well as externally to run in a specific environment. This environment provide the vehicle with stability and resources to create enough energy. This energy moves the vehicle in return.
This whole process is responsible for working of a vehicle. But if the environment is changed, whole process gets flipped.
Coming back to the question, a car cannot be moved under underwater because of the following 2 reasons:
1. Inside Work Cycle: If you submerge a car into a water body, the first thing that will happen is the system failure. First water will destroy all the electronic and electric components inside the car. After frying all instruments, the water will then start entering the engine. The lack of air will stop the combustion process and the water will fill engine and fuel tank. In short everything responsible for moving your car is now responsible for it's "Not Moving"
2. Outside Aerodynamics: A car may look like a sealed airtight container when raining but it's aerodynamic structure cannot hold any kind of water pressure from the outside. The car, if driven underwater, will face serious issues with water leakages. Also the heaviness and surface area of the car will act as the supporting villains.
Finally, as a solution, if you want to drive your car underwater, just make 2 adjustments to your car...
Seal pack your car completely, both from outer layer as well the inside components.
And second, provide air and ventilation support through snorkel, that too a long one indeed.
P.S. - But after doing these changes, the vehicle you made will be called a submarine rather than a car.