Henry Bullen was on the bench for Norwich City Under-18s' clash with Fulham on 3rd December 2022 but did not play. He made his Under-18s debut on 7th January 2023 in goal as Norwich lost 4-3 at Arsenal.

He was announced as part of the 2023 academy scholar intake by Norwich on 7th July 2023.

He signed his first professional contract with Norwich on 11th July 2025. He signed until 2026, with a club option, and featured for both the Under-21s and Under-18s during the 2024/25 season.

Signing his first professional deal, Bullen said: "This is something I'm immensely proud of, I've been at the club since I was nine years old. To have worked my way up through the whole academy is something that me, my family, everyone around me, are really proud of. I've got a season ticket here and I've been watching every game for years, so to have signed a professional contract, it means that little bit more for me."