Gaudette was selected by the Vancouver Canucks in the fifth round (No. 149) of the 2015 NHL Draft after he scored 30 points (13 goals, 17 assists) in 50 games for Cedar Rapids of the United States Hockey League.

Gaudette played the next three seasons at Northeastern University, where he scored 142 points (68 goals, 74 assists) in 116 games. As a junior in 2017-18, he led the NCAA in points (60) and points per game (1.62) and became the first player from Northeastern to win the Hobey Baker Award as the best player in NCAA Division I hockey. Gaudette also won the Hockey East Player of the Year, was named to the America Hockey Coaches Association East First-Team All-American and ended his collegiate career as Northeastern's leader in power-player goals (32).

On March 26, 2018, two days after Northeastern was eliminated from the NCAA Tournament, the Canucks signed Gaudette to a three-year, entry-level contract. He made his NHL debut three days later against the Montreal Canadiens and played five games, going scoreless.

Gaudette earned his first NHL point, an assist, in a 4-1 loss at the Arizona Coyotes on Oct. 25, 2018. He scored his first NHL goal in a 4-2 victory at the Los Angeles Kings on Nov. 24.

Gaudette signed a two-year, $4 million contract with the San Jose Sharks on July 1, 2025, after playing for the Chicago Blackhawks, Ottawa Senators and St. Louis Blues.