After playing two seasons with Jackson Memorial High School and one season with the New Jersey Hitmen of the Empire Junior Hockey League, Stolarz moved to Corpus Christi, Texas, to finish high school and play with the Corpus Christi IceRays of the North American Hockey League.

[2] The Stolarz brothers bonded as children by watching old videos of Brodeur and Ron Hextall on the family television set, with Anthony asking questions about the goaltenders.

He passed on a chance to serve on the second-level team, instead going to Albany, New York, to try out for the Corpus Christi IceRays of the North American Hockey League (NAHL).

[12] After playing only eight games for Omaha, Stolarz dropped out, accepting an offer from the London Knights of the Ontario Hockey League (OHL).

After posting a 13–5 record, 2.53 GAA, and .923 SV% in the first rounds of the 2013 OHL playoffs, Stolarz was benched during the championship series against the Barrie Colts due to a subpar performance.

[13] The January 17 game was Stolarz' 50th regular-season appearance with the Knights, and at the time of his injury, he had a career 34–7–4 record with a 2.46 GAA, .923 SV%, and four shutouts.

[4] Shortly after his return, Stolarz received an eight-game suspension on March 26 after slashing Windsor Spitfire forward Josh Ho-Sang in the back of the head.

[15] Although he missed most of the playoffs, the OHL waived the final two games of Stolarz' suspension, allowing him to play in the 2014 Memorial Cup.

[19][20] Stolarz spent the 2014–15 season with the Lehigh Valley Phantoms of the American Hockey League (AHL), going 9–13–14 with a 3.28 GAA and .905 SV% in 31 games.

[22] On March 2, 2016, Stolarz set a Phantoms franchise record for saves, stopping 51 shots in a 3–2 shootout win over the Syracuse Crunch.

[26] He played seven games with the Flyers in the 2016–17 season, including one shutout against the Detroit Red Wings, and posted a 2–1–1 record with a 2.07 GAA and a .928 SV%.

[27] Stolarz received a qualifying offer from the Flyers on June 25, 2018,[32] and was re-signed to a one-year, two-way contract as a restricted free agent on July 18.

[33] He spent the bulk of the 2018–19 season with the Phantoms, but after a series of injuries befell Neuvirth, Lyon, and Brian Elliott, and Calvin Pickard was traded, Stolarz and rookie Carter Hart became the Flyers' primary goaltenders by January 2019.

[36] Stolarz made his team debut on February 27, 2019, replacing starting goaltender Mikko Koskinen in the second period of a 6–2 loss to the Toronto Maple Leafs.

[42] Stolarz spent most of the 2019–20 season with the Gulls, accumulating a 21–12–5 record, 2.65 GAA, and .922 SV% in 38 games and making an appearance at the AHL All-Star Classic before being called up to the Ducks on March 1, 2020.

[35] On April 12, Stolarz set a Ducks franchise record for most saves by a goalie in a shutout, stopping 46 pucks in a 4–0 win over the San Jose Sharks.

After winning a Stanley Cup with the Panthers, Stolarz left the team as an unrestricted free agent, and signed a two-year, $5 million contract with the Toronto Maple Leafs on July 1, 2024.

[51] Stolarz made his senior international debut at the 2021 IIHF World Championship, starting Game 1 in goal for the United States.