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. 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. 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. 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. In his post match interview, he cited the important roles club mate
Joe Marler and scrum coach
Adam Jones had on influencing his performance. 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. 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. 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. He also started in their semi-final elimination against
Toulouse. Following an impressive season he was named in the Premiership Rugby Team of the Season for the
2023–24 campaign. In October 2024, he scored his first try of the season during a 17–10 victory over
Saracens. ==International career==