McKee began his career on schoolboy terms with NIFL Premiership club Cliftonville, before signing for Stevenage for an undisclosed fee in the summer of 2015. The move came about after he had impressed then-Stevenage coach Darren Sarll who was watching a Northern Ireland U16s match in Belfast. Sarll was there to finalise the signing of Nathan Kerr, but also ended up enquiring about McKee joining Stevenage on scholarship terms.
Stevenage
He played in a number of Stevenage's pre-season friendlies ahead of the 2016–17 season, and subsequently signed his first professional contract in August 2016. He continued to feature predominantly for the club's U18 side during the campaign, and signed "an extended deal" with the Hertfordshire club in February 2017. McKee made his professional debut in April 2017, coming on as a 71st minute substitute in a 4–0 away loss to Colchester United. After the match, he was singled out for praise by Sarll, now the club's manager, who stated "Mark can be anything he wants. He can be the best one we have ever blooded. In all the years back, even extending back to George Boyd, this boy can be a very good player". Two weeks later, on 22 April 2017, he made his first starting appearance, playing the opening 55 minutes as Stevenage fell to a 1–0 defeat to Mansfield Town at Broadhall Way. Following on from his two appearances in the first-team the previous season, McKee opened the 2017–18 season with a place in the starting line-up, playing as a striker in Stevenage's 3–3 draw against Newport County on the opening day of the campaign. He made a sporadic number of appearances in the first three months of the campaign without cementing his place in the first-team. This changed after he was deployed in a midfield role in Stevenage's 1–1 home draw against Coventry City on 21 November 2017, earning Man of the Match for his performance and also being described as possessing "the most amazing awareness and intelligence to his play" by manager Sarll after the match. Stevenage chairman Phil Wallace stated there had been interest from a number of Championship and Premier League clubs for McKee during the 2018 January transfer window, with the club discouraging bids for the player as they believe he will develop faster playing first-team football as opposed to "being lost in the U23 environment of a Premier League club". McKee scored his first professional goal on 3 February 2018, courtesy of a low drilled shot in a 3–2 defeat at Accrington Stanley. He made 29 appearances during the campaign, scoring once, in what was McKee's breakthrough season as a first-team regular. Despite appearing regularly for Stevenage the previous campaign, McKee played just once during the first half of the 2018–19 season. He returned to the first-team as a 79th-minute substitute in a 1–0 home loss to Bury on 9 March 2019. McKee ended the season coming on as a substitute in Stevenage's last four matches as the club finished in tenth position in League Two, one point off of the play-off places. He made eight appearances during the season. McKee was released by Stevenage at the end of the season.
Cliftonville
McKee rejoined his former academy club Cliftonville on 10 October 2019.
