Prior to his professional career, Pospíšil played for the Sioux City Musketeers of the USHL.

[2] On 13 August 2020, the Flames loaned Pospíšil to HC Košice of the Slovak Extraliga.

[4][5] On 3 November 2023, Pospíšil was called up by the Flames and made his NHL debut the next day, scoring his first career NHL goal in a 6–3 win against the Seattle Kraken.

[6][7] On 18 January 2024, the Flames placed Pospíšil on injured reserve after he suffered an upper-body injury in a 4–3 loss to the Toronto Maple Leafs.

[8] On 6 March, Pospíšil was suspended three games for boarding Seattle Kraken's defenceman Vince Dunn.