Kalvach began his career playing at youth level for hometown club Sigma Olomouc.

He was assigned to the reserve squad in 2015, making 26 appearances and scoring twice during the 2015–16 Czech National Football League season.

[citation needed] On 16 May 2019, Kalvach joined Viktoria Plzeň on a three-year deal.

[3] On 14 October 2019, he debuted as a starter in a friendly match against Northern Ireland which ended in a 2-3 loss for the Czechs.

This biographical article related to association football in the Czech Republic, about a midfielder, is a stub.