I’m afraid you’ll have to learn Korean first!
From Jeju’s ‘endangered’ dialect article from The Jeju Weekly :
An after-school program for learning the dialect is offered in a couple of public schools.
This suggests that it should be possible. The article even mentions an iPhone app about the Jeju language, but I didn’t find it.
The Jeju Special Self Governing Province site has a page on the Jeju language (original in Korean, or English via Google Translate). The most useful part for you might be the contact name and phone number at the end.
Credit:stackoverflow.com‘