No water comes out of the toilet’s refill tube

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.

They're also known as vacuum breakers, such as this one. Some searching shows that they're available from Lowe's and HD (assuming you're in the US), but you should be able to find one in a store near you or online if not.