Carson began his football career at Killyleagh Boys before he joined Sunderland in 2004. He then progressed the ranks at the club's youth system. As a result, he signed professional terms with Sunderland and made his debut on 16 July 2006. During his time at Sunderland, Carson was loaned to Chesterfield, Lincoln City, Brentford, Bury, Hull City, and then Bury again. Bury were so impressed by him that, on 22 May 2012 they signed him on a two-year deal and he received the number 1 shirt, and later, was made team Captain. On 3 October 2013, Carson joined Portsmouth in a three-month loan deal, and later left Bury in 2014 and joined Cheltenham Town. In 15 June 2015, Carson signed for Hartlepool United for an undisclosed fee. He kept four consecutive clean sheets between 15 March 2016 and 28 March 2016. In June 2017, Carson joined Motherwell on a three-year contract. Where he remained until he joined Dundee United in July 2021 on a two-year contract. On 5 January 2022, Carson rejoined former Motherwell boss Stephen Robinson after joining EFL League One side Morecambe on loan for the remainder of the season. In May 2022, it was confirmed that Carson would join St Mirren on a two-year deal for an undisclosed fee. In March 2023, he signed a contract extension keeping him there until the end of the 2024/25 season.