Although many people choose to ignore the law a visa IS required to legally do any kind of work in Korea. If you’re caught working without the proper papers you will be fined and risk deportation. This applies to any kind of employment including English tutoring.
Koreans follow the American model of summer camps for kids and such camps are always looking for young people to teach English and join their organisation teams.
The following websites will provide you with a lot of material to get you started: