Yes, but it would probably be a very poor engine.
As you noted in your question, there are huge practical issues with using the arc alone as your power source, compared to using it to ignite the fuel mixture in a conventional spark-ignition engine. But you are correct to observe that both the electric arc and the combustion process cause a rapid expansion inside the chamber, and that this rapid expansion could drive a piston.
What you propose is to convert electrical energy to thermal energy to kinetic energy and useful work (you want to move the car). Implicit in this is some form of energy storage, which we can assume to be chemical for the sake of comparison with conventional technologies. So, chemical -> electrical -> thermal -> mechanical. You have to go through at least three conversions of energy from one form to another in order to make this technology work.
Contrast that with electric motor technology, which stores energy in batteries (or fuel cells) and produces work relatively efficiently. Chemical -> electrical -> mechanical.
Contrast it also with the conventional internal combustion engine, which stores energy relatively densely in hydrocarbon fuels and produces relatively more work per refueling cycle. Chemical -> thermal -> mechanical.
Your proposed solution adds another conversion step to the electric motor solution. It seems to combine the worst characteristics of both established solutions, since you have the same storage challenges to work with as the electric car does. So I wouldn't call it a practical technology; but you might be able to produce a working prototype of this sort of engine, for use as a novelty or teaching aid.