[1][2] Mølgaard was born in Gudum, Lemvig Municipality, Denmark.
As a youth, he started playing football for local clubs Gudum IF, Lemvig GF and Holstebro Boldklub, before joining the Thisted FC youth academy.
[3] He made his senior debut on the last matchday of the 2013–14 Danish 2nd Divisions season under head coach Christian Flindt Bjerg.
[5] On 4 January 2018, it was announced that Vejle Boldklub had signed a two-and-a-half-year with Mølgaard starting from 1 July 2018, after he had made his breakthrough for the Thisted first team.
[3] On 13 June 2022, Danish Superliga club AGF announced they had reached a deal to sign Mølgaard from Vejle on a three-year contract.