Issues that can cause a car to vibrate on acceleration include bad brake pads or disks. The car may be out of alignment, or there may be worn front suspension parts.
If the brake pads or disks are the problem, then replacing them should eliminate the shaking. When you replace one side, replace the other. If the alignment is bad, get the car aligned to make the vibration stop. If a front suspension part has become worn, the exact part must be determined and replaced. These parts include the tie rods, ball joints, steering rack, wheel bearing, strut bearing and struts.