I've noticed that most of the ESL job openings require an English native speaker for the position. Are ESL teaching positions available for non-native speakers?
I've talked about this before in reference to a questioner who was wondering about his Hungarian girlfriend.
...there is very little chance for her to get a standard language education visa (E2) since she is from Hungary. If she were to speak and teach another commonly taught language in Korea, then perhaps she could find a job here, but I'd say her options are limited on that front.
Honestly, the answer is 'no'. You can't teach in the ESL industry in Korea if you are a non-native speaker. If you want to legally teach English or any other language in Korea, you will need an E2 visa. There are other visa options if you are set on coming to Korea, but for English education, you'll need your E2. The requirements for an E2 depend on the language you're teaching, but in the case of English, you must have a valid passport from either US, UK, Ireland, Canada, South Africa, Australia, or New Zealand.
Your other option is traveling to Korea on a tourist visa and trying to find private tutoring gigs, but they often are unreliable and sometimes difficult to obtain. Sorry to deliver bad news.
If anybody has any questions, just send me an email at firstname.lastname@example.org or leave a comment.