Palmer was born in Derby, Derbyshire.[5] He joined Burton Albion's youth academy after being released by Derby County.[6] He secured the award for most improved young player at the end of his first season as a scholar at the club.[6] He made his debut on 15 November 2012 as an 89th-minute substitute in the FA Cup, in a 2–0 away win over Altrincham.[7] He made his League Two debut on 5 April 2013, coming on 79 minutes into a 7–1 away defeat to Port Vale.[7] Palmer's first goal came on 10 November 2013 in a 2–0 home win over Hereford United in the first round of the FA Cup.[8] He came on as a 78th-minute substitute on 26 May in the 2014 League Two play-off final, in which Burton were beaten 1–0 by Fleetwood Town at Wembley Stadium.[9]
Palmer signed for League One club Rotherham United on 25 January 2018 on a two-and-a-half-year contract for an undisclosed fee.[11]
He joined newly relegated League Two club Bradford City on 5 July 2019 on a season-long loan.[12]
Swindon Town
Palmer signed for club Swindon Town on 31 January 2020 on an 18-month deal after his contract was terminated at Rotherham United.[13] He scored his first goal for Swindon in an EFL Trophy tie against Exeter City on 6 October 2020.[14]