Seelt is a youth product of PSV, having joined the club in 2017 from NEC. He signed his first professional contract on 23 May 2019, the day of his 16th birthday. The deal linked him to PSV until 2022. He made his debut for Jong PSV on 11 December 2020, coming on as a substitute in a 3–2 loss to TOP Oss. Seelt would go on to make his PSV senior debut in a 2–1 away win over PEC Zwolle.
On 23 June 2023, EFL Championship club Sunderland announced that it had reached an agreement with PSV for the transfer of Seelt. The deal became subject to a medical and international clearance. The deal was completed on 25 June 2023 with Seelt joining for an undisclosed fee, signing a five-year deal.