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.

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.