[2] After two seasons with Providence College, Stauber opted to turn professional and on March 23, 2022, signed a two-year, entry-level contract with the Chicago Blackhawks.
[3][4] In his NHL debut on January 21, 2023, Stauber made 29 saves in the Blackhawks' 5–3 win over the St. Louis Blues.
[5] Stauber later became the first Blackhawks goaltender to start his NHL career with a 3–0–0 record following subsequent wins over the Calgary Flames and Arizona Coyotes.
[6] During the 2023–24 season, on February 16, 2024, Stauber became the first AHL goaltender to score a goal and record a shutout in the same game en route to the Rockford IceHogs' 4–0 win over the Chicago Wolves.
[7] As a free agent from the Blackhawks, Stauber was signed to a one-year, two-way contract with the Utah Hockey Club on July 5, 2024.