Bonham secured a scholarship deal with Watford in June 2010 and quickly signed his first professional contract three months later. On 12 June 2013, he signed a two-year contract with Brentford. Bonham spent much of the 2013–14 season away on loan at Arlesey Town. Joining on 15 November 2013, he made his debut the following day in a 2–0 FA Trophy third qualifying round victory over Marlow. Bonham helped Arlesey to three league wins in a row in December and at the end of the month, his loan was extended until the end of the 2013–14 season. Following Brentford's promotion to the Championship, Bonham served as backup to David Button through the 2014–15 season. He signed a new contract on 2 January 2015 and celebrated his signing the following day with a start in an FA Cup third round match versus Brighton & Hove Albion, for which he won social media's Man of the Match. Bonham's performances saw him nominated for the PFA League Two Fans' Player of the Month award and he finished the season with 49 appearances. In August 2018, Bonham joined Bristol Rovers on loan until the end of the season. He had an impressive start to his time with the club and was reported on 15 October 2018 to have "the best save percentage of any first-choice goalkeeper in England so far this season, with 91.5%" Bonham finished the season with 46 appearances. Bonham departed Brentford when his contract expired at the end of the season, and signed a two-year contract with Gillingham. In June 2021, Bonham signed a two-year contract with Stoke City and made 28 appearances during the 2022–23 season and extended his contract by another year.