It is suggested to carry small amount of cash while you are in Jeju.
Generally, most shops will take credit cards, except few street ones.
Taxies will take credit cards.
For the Airport Limousine, it is possible but at some locations need to use cash.
* Airport limousine buses available every 15 minutes at the airport (vice versa) and the fare is 4,900 KRW (~ $5 USD).
In short, major credit cards will be accepted in most places, however, at some locations cash might be necessary. Thus carrying small amount of cash in KRW (~ $50 – 100 USD) would be a good idea for small purchases.
* Currency Exchange:
There are currency exchange spots at the airport accepting major currencies. The rates are better there than the street.