A product of the Randers FC youth academy,[1] Bundgaard made his first team debut on 30 October 2019 in the 2–0 win over Viby IF in the Danish Cup.
[2] He made his debut in the domestic league on 7 June 2020, coming on as a substitute in the 83rd minute for Johnny Thomsen in a 2–1 loss to FC Copenhagen.
Upon signing the extension, sporting director Søren Pedersen stated that "[i]t's so great to be able to extend the deal with one of our own guys, who during the spring showed that he can play at the highest level in Denmark".
[4] On 27 October 2021, Bundgaard scored his first professional goal in a 4–0 win in the Danish Cup over Middelfart Boldklub, a strike from inside the penalty ark after an assist by Vincent Onovo.
[7] On 5 June 2021, Bundgaard made his Denmark U20 debut during a 1–0 win over Republic of Ireland in a friendly in Marbella, Spain.