Born in Islington, Shea joined Arsenal's youth team in August 2007. In 2009, he won both the FA Youth Cup and Premier Academy League.
Shea joined Football League One side Southampton on a month-long loan in February 2011 to act as cover for the injured Bartosz Białkowski. However, he was recalled after eight days by Arsenal without making an appearance.
Shea was sent out on loan for the entire 2011–12 season to Football League Two side Dagenham & Redbridge. He made his debut as a substitute for Chris Lewington in the match against Macclesfield at Moss Lane. Despite getting a yellow card for wasting time, he helped his side to win their first league game of the season by scoreline of 1–0.
He was recalled from his loan by Arsenal on 5 December 2011. On 18 September 2012, he was on the bench because of injures to Wojciech Szczęsny and Łukasz Fabiański for a UEFA Champions League match against Montpellier HSC. On 30 June 2013, he was released along with nine other young players at Arsenal.
After a short spell with Needham Market of Division One North, in October 2013 Shea signed for Isthmian League Premier Division side Harrow Borough, where he played 36 games, with 32 of these in the league.
Shea joined League Two side AFC Wimbledon on a free transfer on 4 July 2014.
Shea was Wimbledon's regular goalkeeper during the 2014–15 season and at the start of the 2015–16 season, but was replaced by loan signings when he was injured, and was not able to regain his place. He was on the substitutes' bench for all of the games of the League Two play-offs of 2016 where Wimbledon reached the final and eventually won 2–0 against Plymouth Argyle at Wembley thus securing the club's promotion to League One.
On 3 July 2017, Shea joined Luton Town on a one-year contract.
Shea has not yet represented England at any level but he is also eligible to represent the Republic of Ireland. In August 2010, Shea trained with the England squad at Arsenal's London Colney training ground because of an injury to David Stockdale.Arsenal
Premier Academy League: 2008–09
FA Youth Cup: 2008–09.
2016 Football League Two play-offs: winner's medal