Murray began his junior career in his hometown of St. Albert, spending three seasons with the St. Albert Sabres of the Alberta Major Bantam Hockey League (AMBHL) and St. Albert Flyers of the Alberta Major Midget Hockey League (AMMHL). He then began in Junior A with the Spruce Grove Saints of the Alberta Junior Hockey League (AJHL), leading the team to an AJHL championship in 2015 and winning both the AJHL and CJHL Best Goaltender awards for 2015–16. Murray ultimately joined the Fargo Force of the United States Hockey League (USHL) for his final junior season.
As an undrafted free agent, Murray signed an amateur tryout contract with the Texas Stars of the American Hockey League (AHL) after his final season at UMass,[2] appearing in eight games across the regular season and playoffs. During the following offseason, Texas signed Murray to a full two-year contract.[3]
On October 31, 2022, Murray was signed to a one-year entry-level contract by the Dallas Stars of the National Hockey League (NHL), Texas' major-league affiliate.[4] Shortly afterwards, he was recalled to Dallas' main roster as the result of an injury to starting goaltender Jake Oettinger, with Murray serving as backup to Scott Wedgewood.[5]
After further injuries to Wedgewood, Murray was again recalled to Dallas' roster on February 22, 2023;[6] Murray then made his NHL debut on March 2, earning his first win and saving 19 of 21 shots faced in a 5–2 victory over the Chicago Blackhawks.[7] Murray later signed a one-year extension with Dallas on June 15, 2023.[8]
On January 8, 2024, Murray earned his first NHL shutout in a 4–0 win over the Minnesota Wild.[9]
Murray appeared for the Canada West team in four games at the 2015 World Junior A Challenge, posting a .911 save percentage and 2.75 goals against average en route to a gold medal.