Blackwood was selected by the New Jersey Devils in the second round (No. 42) of the 2015 NHL Draft after he won 33 games and had a .906 save percentage for Barrie of the Ontario Hockey League.

He signed a three-year, entry-level contract with New Jersey on Dec. 30, 2015 and played the 2016-17 season with Albany of the American Hockey League, where he went 17-14-3 with a 2.55 goals-against average and a .907 save percentage in 36 games.

Blackwood struggled in 2017-18 and played five games in the ECHL, where he had a 2.48 GAA and a .920 save percentage. He went 7-17-1 in 32 games with Binghamton of the AHL.

But he responded in 2018-19 and split the season between Binghamton and New Jersey. He made his first NHL start against the Columbus Blue Jackets on Dec. 20 and made 36 saves in a 2-1 loss. He got his first win when he made 40 saves against the Boston Bruins in a 5-2 victory on Dec. 27.

Blackwood got his first NHL shutout in his third start, a 2-0 win against the Carolina Hurricanes on Dec. 29, 2018. He made 37 saves. He then became the first Devils rookie to have back-to-back shutouts when he made 25 saves against the Vancouver Canucks on Dec. 31. He finished the season 10-10-0 with a 2.61 GAA and a .918 save percentage in 23 NHL games.

Blackwood had 22 wins, a .915 save percentage and three shutouts in 43 starts for the Devils when the 2019-20 NHL season was paused March 12 due to concerns surrounding the coronavirus. He was traded to the San Jose Sharks on June 27, 2023, for a sixth-round pick in the 2023 NHL Draft and signed a two-year, $4.7 million contract July 1, 2023. He was traded to the Colorado Avalanche on Dec. 9, 2024, signed a five-year contract Dec. 27, and teamed with Scott Wedgewood to win the 2026 William M. Jennings Trophy as the goalies who played a minimum of 25 games for the team allowing the fewest goals (203 including shootout).

Internationally, Blackwood represented Canada at the 2016 IIHF World Junior Championship.