They made plans to. Roh Moo-Hyun, president of Korea in 2003, announced that the capital would be moved to the city of Gongju in Hoseo. The move was planned to go into effect in 2007, with the construction of the city finished in 2030. This was done both to establish a safe capital, and to lessen the importance of Seoul, which is the economic, population, and political heart of South Korea.

The Supreme Court, however, declared that the movement of the capital by law was unconstitutional, as such an important decision required a national referendum. Though a referendum was never held, polls show a majority of South Koreans opposing the move.

Roh was thus forced to compromise, proposing a new movement of the capital to the city of Sejong, and making it a special administrative city instead of the national capital. Though there was opposition to this move as well, Sejong has served as the de facto administrative capital of South Korea since 2012. 36 government offices have moved here, though crucial portions of the government, including the national assembly, remain in Seoul.

Sources: http://news.bbc.co.uk/1/hi/world/asia-pacific/3554296.stm

http://m.koreatimes.co.kr/phone/news/view.jsp?req_newsidx=114241

Apologies for anything wrong with my formatting; this is my first post on this sub.