Vogt is a product of the youth academies of the Swiss clubs Wohlen, Baden and Aarau. He returned to Wohlen in 2021 in the Swiss 1. Liga and signed his first professional contract with them on 18 July2022 and was promoted to their senior team. He transferred to St. Gallen's reserves in 2023. For the 2024–25 season he scored 13 goals in 25 games with St. Gallen U21, and made 2 appearances with the senior team in the Swiss Super League. On 22 July 2025 he was formally promoted to St. Gallen's senior team and signed a professional contract until 2029.