Does the vibration go away when you are coasting (not accelerating, not braking, just maintaining speed on level ground)?
If so, I wonder if it's rear wheel bearings. Fronts could do it too, but you wouldn't feel them as much when accelerating unless you were in 4WD. When power is applied, your brake rotors and drums are released, and energy surges through the prop shaft in through the diff into the axles, hubs and to the wheels. Usually, a damaged prop shaft, diff or axle won't vibrate when braking because that extra energy isn't there.
The rear bearings are a constant in both acceleration and braking. Jack up the rear of the car and check them out! Has your ABS light come on? The ABS sensor in the rear is near the bearing and a destroyed bearing often gums up the ABS sensor.