In 2015 at the age of thirteen Baxter joined the Harlequins academy and in June 2020 upgraded to full-time basis as a tighthead prop.[1][2] He would later switch permanently to loosehead.[1][3] In December 2020, Baxter made his debut for the first team coming on as a replacement in a defeat against Racing 92 at the Stoop.[4]
In June 2021, Baxter won the Premiership title, with Harlequins beating Exeter Chiefs 40-38 in the highest scoring Premiership final ever. Although he did not play in the final, his appearances as part of the first team throughout the season meant he qualified as part of the squad.[5]
In December 2022, aged only 20 Baxter won Man of the Match for his performance against Racing 92 in the Champions Cup as Harlequins won the match 14-10.[6] In his post match interview, he cited the important roles club mate Joe Marler and scrum coach Adam Jones had on influencing his performance.[7]
During the 2023–24 European Rugby Champions Cup, Harlequins beat Racing 92 again this time at the La Défense Arena as they won 28-31 away.[8] In April 2024, Baxter started as they beat Glasgow Warriors 28-24 at home in the Champions Cup Round of 16, the first time the club had ever won a knockout game in the competition. Despite the victory, he was replaced earlier than usual by Joe Marler after sustaining an injury to his arm.[9] The following week in the absence of Joe Marler, he started against Bordeaux Bègles, winning 42-41, making it only the second time the club had won in the knockout stages of the competition and the first away victory at this stage. His performance was highly praised having won multiple penalties at the scrum against international prop Ben Tameifuna.[10][11] He also started in their semi-final elimination against Toulouse.[12] Following an impressive season he was named in the Premiership Rugby Team of the Season for the 2023–24 campaign.[13]
In May 2024 Baxter received his first call-up to the senior England squad by coach Steve Borthwick for a training camp in preparation for their tour of New Zealand.[19] On 6 July 2024 he made his Test debut coming off the bench as a first-half replacement for the injured Joe Marler in a loss at Forsyth Barr Stadium.[20] The following weekend saw him make his first start as England were defeated by New Zealand at Eden Park to lose the series.[21]
