You wouldn't want to use a .22lr cartridge for suicide even if you HAD a gun, let alone without one. Do you know how guns work? The bullet is only deadly when it builds up enough velocity from the burning powder behind it. That thing you see in the movies/video games when a crate of ammo is blown up and you see bullets wizzing past? Complete fantasy.
Basically, you are going to need a gun otherwise you just have a cartridge with nothing to fire it. You could however manufacture a zip gun if you were truly desperate enough...
I wouldn't recommend it though.