Animal Crossing: City Folk for the Wii has the ability to take in-game screenshots and save them on an SD card in the Wii. Is the screenshot ability strictly dependent on the game, or is it included in the Wii system somewhere? Can I use it to take screenshots of Virtual Console games?
No, you can't.
This is a feature in specific games, not a feature in the Wii itself, so can't be used to screenshot games without this feature.
This is completely game-specific. The Wii has no built-in screenshot ability.