St Mirren Fraser signed a professional contract with St Mirren after coming through the club's youth academy. In July 2022, Taylor made his St Mirren debut coming off the bench in a 3–1 victory over
Edinburgh City in the
Scottish League Cup group stages. In August 2023, Taylor signed a two-year contract extension until summer 2025. After returning from his loan spell with Arbroath, Taylor signed a one-year contract extension with St Mirren in May 2025.
Ballymena United (loan) In August 2023 Taylor joined
NIFL Premiership side
Ballymena United on a season long loan. Taylor won Ballymena's Player of the Year award, after helping the club avoid relegation.
Arbroath (loan) Taylor joined
Scottish League One side
Arbroath on a season long loan in September 2024. Taylor scored his first goal for Arbroath, scoring the winner in a 1–0 home victory over
Stenhousemuir. Following winning the Scottish League One title with Arbroath, Fraser won the PFA Scottish League One Player of the Year for the 2024–2025 season.
Partick Thistle (loan) On 29 January 2026, Taylor joined
Scottish Championship club
Partick Thistle on loan until the end of the season. Taylor made his Thistle debut the following day, coming off the bench in a 2–2 draw with
St Johnstone. ==Personal life==