Annunen first played as a 14-year old with Kiekko-Laser at the junior under-18 level in 2014. Showing promise at age 16, Annunen joined the ranks of Oulun Kärpät, leading the under-18 team in the 2016–17 season, with 30 appearances. In the 2017–18 season, Annunen played predominantly in the Nuorten SM-liiga, posting a 2.21 goals-against average and.907 save percentage. He made his professional and Liiga debut with Kärpät near the conclusion of the regular season, allowing four goals in 33 shots of a shutout defeat to HPK on March 8, 2018. He returned to the Jr. A to be named as the league's best goalie, a first-team all-star and won a silver medal after the club finished in second place. With his draft eligible status, Annunen was ranked as the fourth International goalie by the NHL Central Scouting Bureau and was elevated as the third keeper selected, when he was drafted in the third-round, 64th overall, by the Colorado Avalanche in the 2018 NHL Entry Draft. In order to continue his development, Annunen was loaned by Kärpät to Mestis club Kokkolan Hermes, to begin the 2018–19 season on 31 July 2018. In 22 appearances with Hermes, he had a 12-7-2 record with a 2.77 goals-against average and.891 save percentage, before he returned to Kärpät mirroring his previous season in making one regular season appearance in a 4-3 overtime defeat to KalPa on 14 March 2019. In the post-season, Annunen was assigned to join Karpat's Junior team and posted a 1.50 goals-against average and.930 save percentage in six appearances to help the team capture the Jr. A SM-liiga championship. In the 2019–20 season, Annunen was slated to play in the Liiga to serve as the backup to Patrik Rybár. He collected his first Liiga win in a 3-2 decision in his opening game against KalPa on 12 September 2019. On 16 September 2019, Annunen was signed to an improved two-year contract extension to solidify his role in the Liiga. From September 21 to October 23, Annunen collected four consecutive shutouts to establish a new league record for the longest shutout streak during a single season. Sharing the goaltending duties with Rybár for the regular season leading Kärpät, Annunen posted a 15-5-3 record in 23 games, ranking first among all goaltenders in both save percentage and goals-against average. With the season ended prematurely before the commencement of the playoffs due to the COVID-19 pandemic, Annunen finished the season third in voting for the Urpo Ylönen trophy as the league's best goalkeeper. On 30 April 2020, Annunen was signed by draft club, the Colorado Avalanche, to a three-year entry-level contract. He was assigned to continue his development and tenure with Oulun Kärpät for the 2020–21 season.
International play
Annunen first represented Finland at the 2017 Ivan Hlinka Memorial Tournament before competing at the 2018 IIHF World U18 Championships, the latter of which he helped Finland to a gold medal. He posted a 2.00 goals-against average and.914 save percentage in six games at the under-18 tournament. Annunen represented his country at the 2020 World Junior Championships, where he finished the tournament with a.916 save percentage and a 2.65 goals-against average in six contests. His 362:01 minutes in net was the second-most among all goaltenders in the tournament.
