Chapman signed for Chesterfield from nearby non-leaguers, Belper Town in the summer of 2013. In the 2013–14 season, Chapman spent the second half of the season on loan at Chester, making 16 appearances.
He made his professional debut on 2 September in the Football League Trophy match against Scunthorpe United, after coming off the bench to replace Tommy Lee in the 7th minute.
On 23 September 2014, Chapman joined Accrington Stanley on a two-month loan deal. On 11 October, Chapman had to be substituted in the closing stages of Accrington Stanley's 2–1 home defeat to Dagenham & Redbridge following an injury to his left foot. His loan spell was cut short and he returned to Chesterfield on 13 October, having played three games for Accrington.
On 14 August 2015, Chapman joined Bristol Rovers on a one-month loan deal in an attempt to regain fitness following his injury.
In an EFL Cup tie against Bradford City on 9 August 2016, Chapman came on as a substitute after Elliot Parish was sent off. the game went to a penalty shootout which Stanley won 11–10 and Chapman took and scored a penalty. In an EFL League 2 title winning season 17/18 he won the League 2 Golden Glove award. He was offered a new contract by Accrington at the end of the 2017–18 season.
On 15 May 2018, Chapman joined Peterborough United on a two-year deal to challenge the number 1 Jersey.
He was transfer-listed by Peterborough United at the end of the 2018–19 season.
On 14 December 2019, Chapman joined Tranmere Rovers on a one-week emergency loan.