Undrafted, Kämpf played as a youth within local club, Piráti Chomutov since 2009 at the under-16 level.

[1] He made his senior professional debut in the Czech Extraliga playing with Chomutov, during the 2012–13 season.

On 19 April 2017, despite reported NHL interest Kämpf used his option to re-sign to an additional one-year deal with Chomutov.

[5] As a free agent from the Blackhawks after four seasons with the club, Kämpf agreed to a two-year, $3 million contract to join the Toronto Maple Leafs on 28 July 2021.

[6] Following a productive two seasons with the Maple Leafs, Kämpf as a pending free agent opted to remain in Toronto by agreeing to a four-year, $9.6 million contract extension on 28 June 2023.