A vibrating toilet results from outdated parts inside the toilet's tank. Fortunately you can remove the old parts and install replacements without any special plumbing tools or advanced plumbing knowledge, making the fix a job a homeowner can perform without assistance. The entire job will not take you more than a couple of hours, even if you have never worked on a toilet before.
Ballcock valves, which came with older toilets, use a float that sits on the end of a metal arm. When the metal arm and the float drop all the way to the bottom of the toilet's tank, the ballcock's valve opens up all the way. The valve closes more and more as the float and arm raise, until the float reaches the fill line in the toilet and the ballcock's valve closes completely.
A ballcock's valve uses a rubber seal to keep water from leaking into the toilet when the valve closes. The ballcock connects to a the water supply line that connects to the water valve that comes out of the bathroom's wall. As the ballcock's seal begins to break down and dry out, the seal will cause the toilet to make strange noises when it flushes and even vibrate. The vibrations will start with the ballcock but they will spread through the arm and float, as well as the rest of the tank.
Modern toilets use fill valves, which still use a float and a sealed valve. The float, though, slides up and down along the side of the fill valve. A fill valve will stay open all the way until the float reaches the fill line in the toilet, and then the valve will close all the way, stopping the flow of water. Since the valve stays completely open, it will not emit strange noises and will not lead to the toilet vibrating after you flush.
Updating the Toilet
Older toilets will accept a fill valve, allowing you to swap out the ballcock so you do not have to feel the vibrations or deal with the noises any more. You must close the water valve and then drain the tank using a shop vacuum. With the tank empty, you can unbolt the water line from the bottom of the ballcock on the underside of the tank, and then undo the nut holding the ballcock in place. If the ballcock spins when you try to undo the nut, hold the bottom of the ball inside the tank cock stationary with a wrench. You drop the new fill valve in the toilet and tighten the bolt to hold it in place, as well as bolt the water line onto the bottom of the fill valve.