In his major junior career, Wolf played for the Everett Silvertips in the Western Hockey League (WHL), initially as the backup to Carter Hart before claiming the starting position the following season.

[8] In 2011, Wolf's family moved to Hermosa Beach in Southern California after being persuaded by his agent, and he began to play for the Los Angeles Junior Kings.

[8] Wolf spent seven years with the Junior Kings,[10] including competing in the 2014 Quebec International Pee-Wee Hockey Tournament, where he played against future top CHL prospects Samuel Poulin and Xavier Parent.

[16] Coming in for relief, Wolf made his postseason debut, stopping all 4 shots he faced during 9:22 of ice time before Hart's return at the start of the third period in the 4–0, series-clinching win.

[2] Wolf's numbers improved under his increased workload, posting the highest save percentage and goals against average of any goaltender in the Canadian Hockey League with at least 15 games played, with a .936 SV% and 1.69 GAA respectively, amassing a 41–15–2 record in the process.

[18] On March 28, 2019, Wolf would record his first WHL playoff shutout, saving all 24 shots faced in game 4 of their first round matchup.

[2] Wolf's slide to the final round of the draft was seen as due to his significantly smaller than average stature for modern NHL goaltenders.

Wolf posted stellar numbers for Everett in the pandemic-shortened season, winning 34 out of his 46 games and leading the league in every significant goaltending statistic.

[35] Wolf began the 2020–21 season with the Stockton Heat, making his professional debut on February 21, 2021, allowing 5 goals on 11 shots before being replaced by Garret Sparks in the eventual 7–1 loss to the Toronto Marlies.

[39] Wolf's time in the AHL that season was cut short when his junior team, the Everett Silvertips resumed play in March.

[34] For the 2021–22 season, Wolf would transition to playing professionally full-time, claiming the starting goaltender spot on the Stockton Heat.

[41] Wolf had a standout rookie season with the team, leading the league with 33 wins, posting an eighth-best GAA of 2.35, and the fourth-best save percentage with .924.

[43] During their third-round matchup against the Colorado Eagles, Wolf became the third goaltender in AHL history to record three shutouts in a single playoff series.

[42][49] Before the 2022–23 season the Stockton Heat relocated to become the Calgary Wranglers, principally to allow for players to be called up to the Flames on shorter notice.

[66] Wolf would see slight regressions in his AHL performance this season, placing fourth in save percentage and eleventh in GAA among American league starting goaltenders.

[71][73][74][75] Wolf was recalled by the Calgary Flames on April 12, 2023, where he made his National Hockey League debut alongside rookie Matthew Coronato.

He stopped 23 of 24 shots en route to his first career NHL win, with the Flames defeating the San Jose Sharks 3–1.

[77][78] However, with the Flames already having two goalies in Jacob Markström and Daniel Vladař committed to contracts, they were unable to open a roster spot.

[79] When asked about whether or not Wolf would begin the season with the Flames, general manager Craig Conroy stated "If I was looking right now today, I think the two guys that were here in the years past are really dialed in.

[95][96] Later in the same month, Wolf would start three consecutive games, including earning his first NHL shutout with a 2–0 win over the Nashville Predators on November 15, 2024.

[97][98] Wolf's dominant performances early in the season would spark significant discussion around him potentially winning the Calder Trophy.

[99][100][101] On July 2, 2018, Wolf was named to the USA Hockey National Team Development Program's U-18 roster for the 2018 Hlinka Gretzky Cup.

[104] Wolf would not play in the bronze-medal game, where Team USA would fall to Russia in a 5–4 loss, placing fourth in the tournament.

[107] Wolf played one game in the tournament: a round-robin match against Germany, in which he made 17 saves on 20 shots en route to a 6–3 USA victory.

[110] Wolf's first game at the tournament would be as a mid-game replacement for Spencer Knight on December 25, 2020, stopping all 11 shots faced en route to a 5–3 loss against Russia.

[111] The next night, on December 26, 2020, Wolf recorded his first career World Junior shutout, making 10 saves during Team USA's 11–0 win over Austria.

[112] Through his two appearances at the tournament, Wolf stopped all 21 shots faced and was not credited with a loss, helping lead Team USA towards capturing the gold medal.