Pym came through the youth ranks at his hometown club, Exeter City. Pym was captain of the Under-18 side when he was handed a professional contract by manager Paul Tisdale in April 2013.
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. He went on to become first choice keeper for the last 5 League games of the season and was rewarded with a new contract at the end of the season. He began the season as Exeter's first choice goalkeeper after Artur Krysiak was released.