Stockton came through Tranmere Rovers's youth system. He went on loan to Vauxhall Motors at the end of November 2011. He made his debut as a late substitute, in the club's FA Trophy victory over Marine on 26 November. He scored three times whilst on loan, twice in a 4–4 draw with Kidderminster Harriers, and the winning goal in a 3–2 victory over Worcester City. He made his Football League, debut against Rochdale on 17 January 2012 as a late substitute. He was awarded his first professional contract by Tranmere in April 2012. On 15 August 2012 Stockton scored on his first start for the Prenton Park club against League Two side Chesterfield in the League Cup, a game which Rovers won 2–1. His first Football League goal for the club came on 7 November in a victory over Hartlepool. He scored again against Chesterfield in FA Cup second round as Tranmere knocked Spireites out of the second cup competition of the year. On 29 January Stockton signed a new two and a half-year contract with the Tranmere. With one year left until his contract expired, Stockton extended it until summer 2017 on 26 August 2014. Four days later he scored two goals against Morecambe as Tranmere won the game 2–1. He made ten straight appearances at the start of the season, but later was ruled out of play for two months after picking up an injury before the game against Bury. On 3 January Stockton scored a late goal against Premier League side Swansea City as Tranmere lost 2–6 and were knocked out of FA Cup. He had to miss two months in the spring due to a hernia operation.
Loans
He rejoined Vauxhall Motors on loan on 12 February 2012 before returning to his parent club at the end of the month loan period. In September 2015 he joined Southport for another loan period. In March 2016 he joined Morecambe on loan. In July, he re-joined the club on loan ahead of the 2016–17 season. before returning to Tranmere in January 2017.
Stockton signed for Scottish Premiership club Heart of Midlothian in June 2017. In December 2017, Stockton was criticised for attending a party in blackface, and he subsequently apologised. He failed to score in twelve league appearances for Hearts, and was allowed to leave in January 2018.
Stockton signed for League Two club Carlisle United in January 2018.
Return to Tranmere Rovers
Stockton re-signed for Tranmere Rovers, who were newly promoted to League Two, on 2 August 2018 on a one-year contract after having his contract at Carlisle cancelled by mutual consent. Stockton was loaned out to National League club Wrexham on 7 January 2019 until the end of the 2018–19 season.
