Born in Stirling, McMullan began his career in the under-19 team at Celtic.
Stenhousemuir (loan)
On 23 January 2015, he joined Scottish League One club Stenhousemuir on loan for the remainder of the season. He made his senior debut the following day, playing the full 90 minutes of a 1–1 draw with Ayr United at Ochilview Park, scoring his team's goal. On 14 February, he opened the scoring away to Dunfermline Athletic, who came from behind to defeat Stenhousemuir 3–2. McMullan's only other goal of 14 league appearances came on 11 April in a defeat by the same score at Greenock Morton. Stenhousemuir finished in the relegation play-off places, but stayed up with a 2–1 aggregate victory over Queen's Park: McMullan's free-kick set up Jamie McCormack to head the only goal of the first leg on 13 May, and he also assisted Colin McMenamin in the second leg three days later.
St Mirren (loan)
On 13 July 2015, McMullan was loaned to Scottish Championship club St Mirren for the 2015–16 season. He made his debut 12 days later, playing the full 90 minutes of a 3–1 home win over Berwick Rangers in the first round of the Scottish Challenge Cup. On 7 August, he made his league debut for the team in a defeat by the same score to Rangers at Ibrox. Twenty-two days later, he scored his only goal for the team from Paisley, taking four minutes to open a 1–1 draw against Livingston at St. Mirren Park; the result meant his team had not won yet in four league games of the season. In January 2016, McMullan's loan was ended by Saints, and he returned to Celtic. In total, he played 22 games for the club, scoring once.
Greenock Morton (loan)
McMullan was loaned to Greenock Morton on 24 February 2016, making his debut in a 3–0 Championship win at Hibernian on the same day. On 15 March at Cappielow, he scored his first goal for the Ton, putting them 3–0 up at half time in an eventual 3–2 win over Queen of the South.
Dunfermline Athletic (loan)
On 4 July 2016, McMullan was loaned to newly promoted Championship team Dunfermline Athletic, as PJ Crossan made the move in the other direction on a permanent basis.
Dundee United
On 23 June 2017, McMullan signed a two-year contract with Dundee United.
International
McMullan represented Scotland at the under-16, under-17, under-19 and under-21 levels. He made his debut for the under-21 team in March 2017, appearing in a goalless draw against Estonia.
