Bear was born in Regina, Saskatchewan and raised in the Ochapowace Nation near Whitewood, Saskatchewan and drew inspiration from his brother Everett and other Indigenous hockey players including Carey Price, Jordin Tootoo, Brandon Montour, Arron Asham and Micheal Ferland.

Bear was chosen by the Edmonton Oilers in the fifth round (No. 124) of the 2015 NHL Draft. He played four seasons for Seattle of the Western Hockey League and signed a three-year, entry-level contract with the Oilers on July 2, 2016. He made his NHL debut against the Nashville Predators on March 1, 2018 and scored his first NHL goal in a 5-4 overtime loss to the Anaheim Ducks on March 25.

Bear was sent to Bakersfield of the American Hockey League on Oct. 3, 2019 but was recalled the next day en route to his first full season in the NHL.

The Oilers traded Bear to the Carolina Hurricanes for forward Warren Foegele on July 28, 2021. He had 14 points (five goals, nine assists) in 58 games for Carolina and did not play in the Stanley Cup Playoffs. He and forward Lane Pederson were traded to the Vancouver Canucks on Oct. 28, 2022 for a fifth-round pick in the 2023 NHL Draft.

Bear signed a two-year, $4.125 million contract with the Washington Capitals on Dec. 28, 2023. He was an unrestricted free agent and recovering from shoulder surgery he had in June after aggravating a preexisting injury when he represented Canada at the 2023 IIHF World Championship. The previous season, Bear had 16 points (three goals, 13 assists) in 61 games for the Vancouver Canucks.

Bear signed a one-year contract with the New York Islanders after playing two seasons for the Capitals.