Buxton began his career at Wigan Athletic, and experienced his first taste of first-team football when he was an unused substitute for an FA Cup tie against Hull City on 8 January 2011.[3] Buxton spent the rest of the season playing for the Wigan reserve team, and went on to captain the development squad throughout the 2011–12 and 2012–13 seasons.[2] He broke back into the first-team setup at the beginning of the 2013–14 season, travelling with the senior Wigan side on their pre-season tour of the United States, and featuring in several pre-season friendlies.[2] However, once the newly relegated side's season had started, Buxton found himself unable to make his way into the line-up for the first-team.
Burton Albion (Loan)
On 3 January 2014, League Two side Burton Albion took Buxton on loan from Wigan for one month.[4] He was issued the squad number 4, and made his professional debut on 14 January 2014 in a 4–1 defeat to AFC Bournemouth in the FA Cup.[5] Buxton made no further appearances for Burton, and was recalled by Wigan on 1 February 2014, at the completion of the initial one-month loan.[6]
Accrington Stanley (Loan)
Buxton was sent out on loan again a month later, joining another League Two side Accrington Stanley on 8 March 2014, on a one-month emergency loan deal.[7] He made his debut for the club the same day, starting in a 3–1 league win against Chesterfield. Accrington manager James Beattie said of Buxton's debut performance after the match: "I was very pleased with Adam's display at right-back on Saturday and I think everyone who was at the game saw how accomplished he was."[8] On 10 April 2014, Buxton's loan at Stanley was extended until the end of the 2013–14 season.[9] At the end of the season, Buxton was among five players to be released.[10]
Accrington Stanley
After being released by Wigan Athletic, Buxton re-signed for Accrington Stanley on a one-year contract.[11] He scored his first goal for the club in a 2–1 loss at AFC Wimbledon on 24 January 2015.[12] In August 2016 Buxton declined a contract extension with Accrington in favour of joining Portsmouth.[13] He mutually agreed to leave the club after struggling to make an impact and only played cup matches in his short stay.[14]
Tranmere Rovers
On 1 February 2017, Tranmere Rovers signed Buxton on a free transfer on a deal until the end of the season.[15]
Morecambe
On 29 June 2019, Buxton signed for League Two club Morecambe on an initial one-year deal.[16]
Buxton scored his first goal for Morecambe on 28 September 2019, coming off the bench against Northampton Town to score Morecambe's first of their comeback in a 2–2 draw.[17] Buxton was made available to find a new club alongside teammate Andrew Tutte in January due to limited first team opportunities[18] and was released in June 2020.[19]
York City
Buxton joined National League North side York City on loan until the end of the 2019/20 season.[20]