Gadegaard joined AGF from the local club, Stavtrup IF, at the age of 12 and joined the club's under-13 team.[2] He progressed through the club's youth academy and already at the age of 16, in early 2018, he started training with AGF's Danish Superliga team.[3] On 1 April 2018, he was on the bench for the first time in the Danish Superliga for a game against Hobro IK.[4][5] In May 2019, he signed his first ever professional contract, signing a deal until June 2022.[6] However, in the following two seasons, Gadegaard primarily played for the under-19s and AGF's reserve team.
In February 2021, Gadegaard once again extended his contract, this time until June 2023.[7] In the following 2021–22 season, Gadegaard became starter Jesper Hansen's backup. He also made his official debut that same season when he started in a Danish Superliga match against Nordsjælland on 21 May 2022, which ended in a 2–2 draw.[8]
Esbjerg fB
In search of more playing time, it was confirmed on 29 August 2022, that Gadegaard moved to the recently relegated Danish 2nd Division club Esbjerg fB on a deal until June 2025.[9][10] On 11 September 2022, he made his debut in a match against HIK.[11] Gadegaard also played in the next two games before suffering an injury and sitting on the bench for the rest of the season.
In the 2023–24 season, Gadegaard made seven appearances—two in the Danish Cup and five in the league, featuring in the final five games of the season. He was part of the squad that secured promotion to the Danish 1st Division. Following his strong performances at the end of the season, Gadegaard began the 2024–25 season as Esbjerg's first-choice goalkeeper in the Danish 1st Division and was also assigned the number 1 shirt.[12]