The Faith Militant isn't particularly analogous to any group in medieval Europe. Although the Church had the Inquisition, it was concerned with prosecuting heresy, not general conduct. The individual pastoral figures like priests would have, should a prostitute or brothel owner have visited him, handed out penitence, presumably under the assumption that they were renouncing their sin. But that isn't societal punishment, it is an individual spiritual act. Prostitues also weren't supposed to be part of the church or to donate money to it, but that was sometimes debated within the church. Cities and small towns would have been responsible for policing and punishing people committing crimes. But whether prostitution/brothels were illegal was a matter of local law. There were some jurisdictions, such as Strasbourg, in which it was legal and regulated by local laws.

So, really, there is no strong precedent for a group of medieval fanatics arresting all of the prostitutes and letting the brothel owner go. Religious fanaticism aside, there is some evidence that male brothel owners were exploitative of female prostitutes and that female prostitutes bore most of the stigma while their pimps and johns were largely ignored.

You mention religious fanaticism during the Black Death specifically. What that looked like was not so much targeting the immoral people in the community, but turning inward to expunge personal sin (the flagellants) and encourage others to do so as well, or a turning outward against outsiders such as Jews, Lepers, foreigners, etc, blaming them for poisoning (literally or metaphorically) the community.