Although there is no passage directly in the Torah, many Jewish people cover their heads and it is considered a part of Jewish law. Orthodox men in particular cover their heads all the time but many other denominations choose to cover their heads minimally, only while praying, studying the Torah and when in a synagogue or not at all.
One common head covering is the yarmulke or kippah. These can be any color. Both men and boys wear head coverings including young children.
Typically the characters in Fiddler on the Roof are portrayed wearing an array of different hats and few of them wear the traditional yarmulke with the exception of Mendel, the Rabbi's son.
However, all male Jewish characters (including actors not identifying as male, but playing male characters) should be wearing a head covering such as a cap, hat or yarmulke, at all times.
Appell, V. S. (n.d.) Why do Jews wear head coverings? Retrieved from: https://reformjudaism.org/learning/answers-jewish-questions/why-do-jews-wear-head-coverings
Rozovsky, L. (n.d.) What Is a Kippah (Yarmulke)? Retrieved from: https://www.chabad.org/library/article_cdo/aid/607780/jewish/What-Is-a-Kippah-Yarmulke.htm