[10] Bundgaard made his European debut on 19 August 2021, coming off the bench in the 76th minute for Vito Hammershøy-Mistrati in a 1–1 draw in the UEFA Europa League qualifier against Turkish club Galatasaray.
[21] Bundgaard made his debut for Denmark under-16s on 26 November 2019, starting in a 1–1 friendly draw against Portugal in Antalya, Turkey, under national team coach Søren Hermansen.
[26] He scored his first international goal on 28 March 2022 for the under-18s in a friendly against England, his shot squirming through the arms of goalkeeper Max Thompson shortly before the break in an eventual 3–3 draw.
[28] He immediately scored on his debut against Georgia on 21 September, beating Sporting CP goalkeeper Papuna Beruashvili one-on-one in the first half, as Denmark won 3–1.
He has been noted for his technical ability and creativity, frequently contributing to his team's offensive play by setting up goalscoring opportunities with precise passes and key assists.
Bundgaard is recognised for his high work rate, agility, and proficiency in dribbling, which allows him to navigate tight spaces and challenge defenders effectively.
Although he was not a prolific goalscorer early on, his dynamic movement and ability to drive the ball forward are central to his role in creating attacking chances.