No, think about it. Stored energy is only needed upon start up. Then, you have a resource for an alternate energy and that is motion.
That's not true. The first law of thermodynamics says that energy is neither created nor destroyed. The motion of the car is work, and doing work requires using energy. You cannot do work and generate more energy from the same energy. Doing so would be creating more energy. In order to make a car run, you have to convert energy from some stored form into the motion of the car.