Generally speaking, yes. While parents had lots of power over their children's marriages, agreeing to one selected by those involved would've been possible.

The matchmaker was very much a real thing. While they weren't necessary to set couples up (parents could arrange things themselves), the matchmaker, or shadkhan, would work with the families to find suitable matches.

An important point in this is that a shadkhan or parents setting up a match wasn't the end of the process. Marrying people sight-unseen, arranged by parents and matchmakers, wasn't terribly common. That was the start of the process. After arranging a match, the couple would meet to be sure they like each other, as would their families. Matches can "fall through" at this point in the process.

Keep in mind that Fiddler on the Roof is only a generation removed from Eastern Europe. More importantly, it's based on a short story series called "Tevye the Milkman" by Yiddish writer Shalom Aleikhem, who wrote it in the time and place it's from.