After being cut from his local Kenai River Brown Bears, Swayman played one season of junior ice hockey with the Pikes Peak Miners.

Swayman spent the first part of the season with the Providence Bruins of the AHL but was promoted to the NHL in April after Boston's primary goaltenders were sidelined by injury and illness.

His performance in the remainder of the season led Bruce Cassidy to name Swayman Tuukka Rask's backup goaltender in the 2021 Stanley Cup playoffs, but he appeared in only one postseason game.

After Rask and former backup goalie Jaroslav Halák left the Bruins in 2021 and 2022 respectively, Swayman formed a goaltending platoon with veteran Linus Ullmark, with whom he won the William M. Jennings Trophy for allowing the fewest goals in the 2022–23 season.

[8] He had expected to stay in Alaska and play for the Kenai River Brown Bears of the North American Hockey League,[9] but he was cut from the team in favor of two older goaltenders.

[11] After one season with the Miners, Swayman was selected by the Sioux Falls Stampede of the United States Hockey League in the 12th round of the 2016 USHL Phase II Draft.

[12] Sioux Falls coach Scott Owens had first taken notice of Swayman during his season in Pikes Peak, and he remained impressed by the goaltender during training camp.

After leading all USHL goaltenders in preseason save percentage, Swayman stopped 48 shots on goal in his Stampede debut, a 3–2 shootout win over the Tri-City Storm.

[11] In 32 games with Sioux Falls, Swayman posted a 7–18–3 record, .914 SV%, and 2.90 GAA, and the NHL Central Scouting Bureau named him the No.

[14] After making 40 saves to give the Black Bears a 5–2 win over Boston University on November 17, the Hockey East conference named Swayman their Rookie of the Week.

[17] Swayman finished his freshman season with a .920 save percentage (SV%) and 2.74 goals against average (GAA), while his 15–13–3 record was the best of any Maine rookie goaltender since Ben Bishop.

[18] Swayman's first Hockey East honor of the 2018–19 season came on November 19, when he was named the Defensive Player of the Week after making 79 saves in a two-game series against Boston University.

He earned a number of accolades at the end of the season, beginning with the Walter Brown Award, given to the top American-born college hockey player in New England.

[32] He was the runner-up for the Hobey Baker Award, given to the top men's college ice hockey player in the United States, an honor which went that year to Scott Perunovich of the Minnesota Duluth Bulldogs.

He finished his college career as Maine's all-time leader with 3,130 saves and 5,906:45 minutes played, while his 2.51 GAA was fifth in franchise history and his .927 SV% was behind only Jimmy Howard.

[40] With Tuukka Rask sidelined by an upper-body injury and Jaroslav Halák testing positive for COVID-19, Swayman was promoted to make his NHL debut on April 6, 2021.

His regular-season performance prompted coach Bruce Cassidy to select Swayman over Halák as Rask's backup goaltender in the 2021 Stanley Cup playoffs.

[45] While Halák left the Bruins to join the Vancouver Canucks during the 2021 offseason, Rask underwent surgery to repair a torn hip labrum and was expected to miss the first half of the 2021–22 season.

"[61] Much recrimination ensued about coach Jim Montgomery's roster decisions, in particular the choice to depart from the team's regular season goalie rotation in favor of playing Ullmark exclusively, despite him dealing with an injury.

With the help of Swayman, the Bruins finished second in the Atlantic Division, setting up a first-round match with the Toronto Maple Leafs in the 2024 Stanley Cup playoffs.

[71] As the third goaltender behind Jake Oettinger and Joseph Woll, Swayman received limited playing time in the tournament outside of an exhibition game against Sweden and the bronze-medal match against the Czech Republic.

[72] Swayman played in the final 3 minutes and 19 seconds of the third-place game, allowing no goals as Team USA defeated the Czech Republic 9–3 to take the bronze medal.

[73] Following an injury to Alex Nedeljkovic of the Detroit Red Wings, Swayman joined the United States national team for the 2022 IIHF World Championship in Finland.