NSLI-Y is a fully-funded option for Korea. Applications will come out in the fall prior; it’s become pretty competitive the last few years.

There are many paid options for Japan — AFS and YFU have been around for a while, but there are surely many others.

Here’s a program that looks interesting -

http://www.highschooldiplomats.com/japan

What about something like EF? They offer a language program in South Korea.

I don't know of any programs that take HS kids in Korea. Yonsei only takes graduating seniors and college age kids. I think Ewha Women's university also only takes college kids.

My hs sophomore ds is interested in spending the summer after his junior year in Japan or South Korea on some sort of 'exchange' program. Does anyone have any experience with a program like this? His school is not really a big help.