Among the Ashkenaz Jews, it's known that around 1000 AD, Rabbi Gershom Ben Judah banned polygamy. Probably because the Jews of his time became more wealthy, and more concerned about their daughters' rights. This is one theory, anyway, by Prof. Abraham Grossman. Another theory suggests that Jewish merchants took themselves wives in different towns, leaving their first wives with no support, and therefore this ban came to life. It makes sense because he also banned husbands who left their home for more than 18 months. Not all Jews followed these rules and even today Yemenite Jews can still marry more than one wife.
I can elaborate more when I get home and have access to my books, but...
1 Timothy 3:2
An overseer, then, must be above reproach, the husband of one wife, temperate, prudent, respectable, hospitable, able to teach
Church Tradition states that the Apostle Paul wrote 1 Timothy to his disciple, Timothy, in the first century (60s). It may have been written later and not by Paul, depending on who you ask. That's not important to your question, beyond the fact that it could have been written as late as the mid-2nd century CE.
Verse 3:2 outlines the qualifications for an overseer (or bishop), which is a position of leadership within the Christian Church. It appears to very clearly prohibit church leaders from having multiple wives. This understanding can be backed up with the teachings of early church leaders.
John Chrysostom (c.349 - 407), an Archbishop of Constantinople, reinforces this qualification in his homilies on 1 Timothy. He states:
“A Bishop [overseer] then,” he says, “must be blameless, the husband of one wife.” This he does not lay down as a rule, as if he must not be without one, but as prohibiting his having more than one"
So, John Chrysostom taught that marriage itself was not a requirement for an overseer/bishop, but only having one wife is a requirement.
Note: this only concerns bishops, not laity. To my knowledge, John Chrysostom does not address the number of wives a man outside of the clergy can have. However, the language he uses indicates that his teachings are directed to households with a single husband and wife. Someone with more knowledge about early church teachings can chime in on that bit.
Conclusion: prohibiting church leaders from having multiple wives could have arisen as early as 60 CE. At least in Constantinople (one of the most important centers of Christianity until it's fall in the 1453), it was definitely prohibited by the year 400 CE, if not earlier.
Sources:
1 Timothy 3:2, New American Standard Version
The Church Fathers; Philip Schaff (2014-06-12). The Complete Ante-Nicene & Nicene and Post-Nicene Church Fathers Collection: 3 Series, 37 Volumes, 65 Authors, 500 Books, 18,000 Chapters, 16.5 Million Words (Kindle Locations 441725-441726). Catholic Way Publishing. Kindle Edition.
Edited to for clarity