Please note that the list of resources here is not meant to be exhaustive. You can use this information as a starting point for researching schools that fit your own personal situation and needs.
Homeschooling is not common in Korea. All Korean citizens are required to be educated in the school system from Grades 1-9. The ministry of education does not officially recognize homeschooling as a schooling option, so there is not much of a support system for homeschooled students. So in principle, not sending your child to school is illegal. However, no one has ever been accused as guilty for this yet. There is a growing number of homeschooling families in Korea for various reasons. Some (Christian) families choose to homeschool their children due to religious values and some homeschool their children to avoid different conflicts at school.
Homeschooling requires serious commitment and unique lifestyle as a family. Careful consideration is needed before anyone chooses to take this schooling route.
This is perhaps one of the most organized support system for homeschooling families. This "academy" is run by a church, and is staffed by a dedicated ministry team. There is are regular meet-ups where you learn from a Biblical curriculum, go over academic subjects together, and participate in extracurricular activities such as orchestra. They also host a homeschool expo, where various homeschooling resources are exhibited. Most students at GHSA follow the Korean school curriculum.
There are several other websites and co-ops related to homeschooling. However, they are mostly in Korean, and the sites are not very active or information is not very up-to-date. Some are listed below, and you can explore them on your own.
This program is meant for high school students studying the US school curriculum.
Taking the SAT and other Standardized Tests as a homeschool student
If you are a high school student, one of the greatest challenges of doing online or homeschooling in Korea is finding standardized testing centers open to outside students.
SAT test registration is relatively straightforward. You can just go to the College Board's website and find open test centers. Note that seats fill up fast, so it is best to sign up for the tests in late summer when the sign ups become available for the academic year.
For ACT and AP tests, Fulbright Korea provides registration and proctoring services. Fulbright Korea does not offer AP tests in all subjects but tends to cover subjects that are popular among Korean students. For example, they do not offer AP English though this might change. If you would like to take AP tests that are not offered by Fulbright Korea or prefer to take the test in a different location, you should call test centers on your own and ask if they accommodate test takers outside of their institution. Seoul Christian Academy in 봉천동 has been very accommodating such needs.
For 검정고시, or middle- or high-school graduate equivalent tests, you can find information here. If you are a home school student and would like to apply to a Korean university, it is mandatory to take this test. It is given twice a year. Note that any extra-curricular or academic achievements records established after the test data does not count towards your college application since technically they are achievements after graduation. So be careful in planning the timing of the test taking.
Going to college as a home school student
If you are going to go to college in the US, the role of parents as the head of school and a counselor is important. You can submit transcripts issued by your parent, and do not necessarily have to take the GED or equivalent test to be officially recognized as a high-school graduate or the equivalent. Key elements for a successful college application would be providing convincing and objective documents showing your level of academic accomplishments. Some examples include high standardized scores such as SAT, ACT or AP tests to show that your scores are commensurate with the GPA issued by your home school transcripts. It is also important to submit letters of references who have taught you or interacted with you. A teacher from an online class, for example, would work for this purpose. Most importantly, show convincing evidence why you chose to homeschool and how you took advantage of this opportunity or overcame the situation to mature and grow. If you're diligent enough, there are enough information on the web and there are even paid counselors who can help you navigate this process. Successful homeschoolers are known to do well in college, so don't feel you're a weirdo. We know of stories of students who struggled in a Korean school yet made it to top US colleges.
If you are going to apply to colleges in Korea, you need to take 검정고시, which is offered twice a year. You will still need to submit SAT, ACT or AP test scores to demonstrate your level of academic achievements. Each university has a different way of evaluating students that are homeschooled, so you should check with the homepages of the admissions office (입학처) in each college. A gradually more number of students are being admitted to top Korean universities including Seoul National University. International colleges at Yonsei, Korea and other university have a similar admission system to US colleges. Some other reputable universities do not require test scores but only look at your English scores and will admit you based on interviews.
You may also consider universities in Songdo, Incheon. Currently there are SUNY Korea, George Mason, University of Utah, and Ghent University there, and they admit students twice a year. They are an international branch of the parent university, and you can spend a year in their main campuses. The teaching is in English and there are some international students.