After Moser developed through the Arizona Coyotes organization, he was traded to the Tampa Bay Lightning by the Utah Hockey Club with Conor Geekie and two picks in the NHL Draft for Mikhail Sergachev on June 29, 2024. He signed a two-year, $6.75 million contract with the Lightning on July 11, 2024, and an eight-year $54 million contract (average annual value of $6.75 million) with Tampa Bay on Dec. 27, 2025.

Moser played five games for Team Switzerland at the 2026 Winter Olympics (one goal, three assists). His goal at 12:48 of overtime gave the Lightning a 3-2 win against the Montreal Canadiens in Game 2 of the Eastern Conference First Round that made him the first player in team history to have his first Stanley Cup Playoff goal be an OT winner.

The native of Biel/Bienne, Switzerland, was chosen in the second round (No. 60) of the 2021 NHL Draft and played 205 regular-season games for the Coyotes. As captain of EHC Biel in Switzerland's National League, the top professional ice hockey league in the country, Moser had 30 points (nine goals, 21 assists) in 48 games of the 2020-21 season. He played 18 games for Tucson of the American Hockey League in 2021-22 before getting promoted to the NHL.

Moser also represented Switzerland at the IIHF World Junior Championship (2019, 2020) and IIHF World Championship (2019, 2021, 2022).