IIRC, none of the Gameboys have region locks. You may need a different power adapter to work with whatever the local current/voltage/prongs are, but the games will work on any Gameboy worldwide (assuming your cousins can understand English, or you’re getting games where reading skills aren’t an issue).
They’re definitely readily available at Yongsan Station in Seoul; a place called Space Nine (yep, that’s even the Korean name for it, pronounced just like in English).