Pym was born in Exeter, Devon.[1] He came through the youth ranks at his hometown club, Exeter City. Pym was captain of the under-18 team when he was handed a professional contract by manager Paul Tisdale in April 2013.[4]
Pym made his debut for Exeter as cover for Artur Krysiak in a 3–2 victory over Southend United at Roots Hall on 15 February 2014.[5] He went on to become first choice keeper for the last five league matches of the season and was rewarded with a new contract at the end of the season.[6] He began the season as Exeter's first-choice goalkeeper after Artur Krysiak was released.[7]
In 2014–15, Pym battled for starting position against fellow 19-year-old James Hamon, eventually playing 25 league games to the latter's 21.[8] The following season, new Austrian signing Bobby Olejnik was the undisputed starting goalkeeper in league games, until Pym edged back into the line-up in October 2016.[9]
In May 2019, Exeter manager Matt Taylor said he expected Pym to leave the club on a free transfer.[10]
Pym signed for League Two club Stevenage on loan on 27 December 2021, effective from 3 January 2022, for the remainder of the 2021–22 season.[12] The move meant Pym would be reunited with manager Paul Tisdale.[12]
On 12 July 2022, Pym signed for Mansfield Town on a season-long loan.[13]