Born in Columbus, Ohio, Sillinger was selected by his hometown Blue Jackets with the No. 12 pick in the 2021 NHL Draft. His dad, Mike, played for 12 NHL teams, including two seasons with the Blue Jackets from 2001-03.

Prior to being selected by Columbus, Sillinger scored 53 points (22 goals, 31 assists) in 48 games with Medicine Hat of the Western Hockey League in 2019-20. When the WHL delayed the start of its 2020-21 season due to concerns surrounding the coronavirus, Sillinger decided to move Sioux Falls of the United States Hockey League

Sillinger made the most of his opportunity by scoring 46 points (24 goals, 22 assists) in 31 games and being named the league's rookie of the year. He had 15 goals and 31 points in his rookie season with the Blue Jackets in 2021-22, which included his first NHL hat trick when the Blue Jackets defeated the Vegas Golden Knights 6-4 at Nationwide Arena on March 13, 2022. It made him the second-youngest player in team history (18 years, 301 days) with at least three goals in one game behind Nikita Filatov (18 years, 230 days) on Jan. 10, 2009.

After Sillinger had 11 points in 64 games the following season, he bounced back with 13 goals and 32 points in 2023-24 before signing a two-year, $4.5 million contract as a restricted free agent Aug. 21. He signed a three-year, $13.875 million contract (average annual value of $4.625 million) with the Blue Jackets on July 25, 2026, avoiding an arbitration hearing. At the time of the deal, Sillinger had at least 30 points in four of his five NHL seasons, and led all players from his draft class in games played (367) and hits (508).