Plumbing – use a 1/4 turn valve in place of multi turn valve for toilet

First, a silly question for you. Are you sure the valve is not a 1/4 turn ball valve, and are you sure it has not turned off the water supply to the tank. If the answer is no to both questions, then we have to assume the valve is defective and we go to plan two. Now you have two jobs ahead. First, replace defective valve with a 1/4 turn ball valve and second, replace the wax ring with rubber flange. Before you can attack the valve replacement, you must turn off the main water supply, usually found at meter or pressure tank. After you have turned off the water supply. open a couple of faucets that are below and above the valve you want to replace. This will drain water from the system and avoid a lot of water held in pipes above the toilet valve from rushing out all over the place when you unsolder or cut the old valve out. Without seeing exactly what configuration your valve is attached by, I cannot advise you if it can be unsoldered. Another option may be to secure the body of the valve with a wrench and see if you can remove the cartridge retaining nut with a separate wrench. This will avoid the fear of ruining the copper in the wall. you may be able to replace just the guts instead of the entire valve. A pic of your set up would be helpful.

I'm guessing the valve assembly inside the toilet is finding a resonance when it's just about to close. You can replace it, and it's not too difficult of a DIY job. They sell just the valves:

But if you're going to take it apart to replace the valve, I tend to buy the full kit and replace all the guts.

Note that when my toilets started to leak, it was because of high pressure in the water lines. So it wouldn't hurt to pickup a pressure gauge that screws onto a hose faucet to make sure you don't have a bigger issue. After I fixed one, another would start leaking next as the pressure got a little higher.