He played with the Charlottetown Islanders until his final year of major junior hockey when he was traded to the Val-d'Or Foreurs.

[2] He went on to capture the 2014 President's Cup with the Foreurs and was recognized for his outstanding play when he was awarded the Guy Lafleur Trophy as the QMJHL Playoffs Most Valuable Player.

[6] He attended training camp in 2014 and had a successful rookie tournament,[7] but was demoted to the teams' American Hockey League (AHL) affiliate, the Toronto Marlies, on September 29.

[17] At the completion of his entry-level contract with the Maple Leafs, Bibeau was not tendered a qualifying offer as a restricted free agent on June 26, 2017.

In his Eagles debut on October 4, 2019, Bibeau allowed five goals in an opening night defeat to the Stockton Heat, while also suffering a hip injury which placed him on the injured reserve.

[24] On November 16, 2019, he became the fourth goaltender to start a game with the Avalanche and also record a win in backstopping Colorado to a 5–4 overtime decision over the Vancouver Canucks.

[28] Bibeau never featured with the Hurricanes in the shortened 2020–21 season, posting 5 wins through 8 games in his assignment to the Chicago Wolves of the AHL.

[29] On August 20, 2021, Bibeau signed as a free agent to a one-year, two-way contract with expansion club, the Seattle Kraken.

[30] Bibeau was assigned to Ottawa's AHL affiliate, the Belleville Senators and was paired with Mads Søgaard as the goaltending duo for the team to start the season.

[31] On November 26, 2022, Bibeau set a career high in saves in a single game, making 49 of them in a 4–1 win over the Laval Rocket, which also became the Belleville Senators' franchise record.