Lee Bum-young signed a professional contract for Busan I'Park in 2008, later making a total of 17 appearances in his first year for the club. After five years of playing in the K-League, he then became the first-choice goalkeeper for Busan. Lee Bum-young spent three years as a first-choice until the club finished 11th in the league in 2015 and were relegated in the playoffs, losing 0-3 on aggregate to Suwon FC. Lee had made a total of 134 appearances in the league and 163 overall for Busan I'Park.
Lee Bum-young then transferred to the newly promoted Avispa Fukuoka, where he was made the first-choice goalkeeper by coach Masami Ihara.
In the 2012 London Olympics, Lee played a crucial role in South Korea's quarterfinal victory over Great Britain. After entering the match as a late substitute for injured first-choice goalkeeper Jung Sung-ryong, he saved Great Britain's fifth kick from Daniel Sturridge in the penalty shootout. Korea scored its fifth and deciding kick and won the shootout 5-4, progressing to the semifinals of the Olympic tournament for the first time in its history. He started the semi-final match against Brazil.
Lee was included on his home team's 2014 World Cup squad as the third-choice goalkeeper. However, he did not play a single minute. He made his senior debut against Uruguay on September 8, 2014.