In January 2019, Obbekjær went on training camp in Turkey with the OB first team. On 1 April 2019 it was confirmed, that 16-year old Obbekjær had signed his first professional contract until the summer 2021 and was promoted permanently to the first team squad.[2] At the end of the month, Obbekjær got his professional debut for OB against F.C. Copenhagen. Obbekjær started on the bench but replaced Mathias Greve in the 64th minute which made him the third youngest debutant in the club's history.[3] In October 2019, Obbekjær became the youngest Danish Superliga starter in OB's history at the age of 17 years and 94 days.[4]
On 22 January 2021, Obbekjær joined Italian Serie B side S.P.A.L. on loan for the rest of the season, with an option to buy.[6] He was registered for the clubs Primavera (U19) team.[7]
After returning to OB, Obbekjær didn't feature on OB's first team list. In July 2021, he went on a trial at Hobro IK, however, he returned without being offered a contract.[8] With Obbekjær's contract with OB expiring at the end of 2021, he was sent to the U19, as he was not registered for the first team.[9]
In June 2022, he joined 07 Vestur in the Faroe Islands.[11] After a good time in Vestur, where he became the club's captain while also making the team of the year in the 2023 season, Obbekjær left the club when his contract expired at the end of October 2023.[12]