It isn't safe to continue using a car when you notice it shakes when you accelerate. Damaged engine mounts can cause your vehicle to vibrate when accelerating. The engine mounts are responsible for absorbing vibration made by the cranking engine, so you are most likely to feel the shaking when they are damaged. Bent driveshaft and axles, and wheel bearing failure, amongst others, can cause your car to shake while accelerating. These car problems can cause severe engine damage if not worked on. It can also result in different internal and external problems, so it is essential to visit a mechanic to resolve the issue immediately.