Recently, I’ve been reading about how there were several competing movements to early Christianity, and that the late Roman Empire was a hotbed for new religions. I then remembered that the United States had some periods of proliferative religiosity itself. While 19th century American culture is obviously its own unique thing, it still struck me as interesting from a sociological perspective that a radically new theology emerged at this time. The fact that it took off probably meant it fulfilled some need for its followers.
I wonder, were there other prophets leading movements that could be compared to Mormonism at this time?
In short, yes. Competitors isn’t quite the word, but does capture quite a bit of the spirit at the time.
Mormonism and it’s contemporary religious branches came out of a movement in American history called the second great awakening in the early 1800s. Like the first great awakening a few decades prior, it was a time of heightened religious awareness and zeal. In the case of the second great awakening, much of it was in reaction to developments coinciding with the growing American republic, such as expansionism, consumerism, the shift to market capitalism, and others. In many cases, these new religious thinkers (many of whom did not start new religions as explicitly as the church of Latter Day Saints) while trying to reject these ideas, also came to embody them in new ways as well. This has led to some historians and modern day theologians referring to the LDS church and Mormonism as a whole as “American exceptionalism, as a religion.” (Not a direct quote).
More to the point of your question, other groups did arise at the time, such as jehovahs witnesses and many black religious groups in America. This was because the second great awakening coincided with other social events at the time that fed into each other and helped to develop the growing abolitionist movement later on. It’s also important to remember that not all Mormons are LDS. There were Mormon groups that broke off from one another, especially as the groups were pushed further and further westward. These often did compete with one another.
Another instance of religious zeal is that of Matthias. Much of his ‘lore’ and religious teachings were almost exact stolen copies of the events surrounding the development of Mormonism. A really interesting read on this subject is “The Kingdom of Matthias” by Paul E. Johnson and Sean Wilentz. It resembled more of an early cult than a religious but is a story that seems wild and dramatic at every turn.
You are definitely correct that the 19th century United States saw its fair share of religious zeal. During this period there was the Second Great Awakening, religious revival camps, travelling evangelists, the advent of pre and post-millennialism, and the glorious burned over district, among many other religious activities of importance that I'm not mentioning.
To start off with, Mormonism specifically was founded in an area of western New York during the early 1800s where there was so much religious fervor that it was termed the "Burned Over District." It was here that Joseph Smith had visions from the divine and founded a new denomination of Christianity.
Now, this aspect of having a divine experience and gaining a following to practice what you're preaching and forming a new denomination around those beliefs was not new in America. Nor was it unusual for these new fledgling denominations to be founded by some charismatic leader. The Burned Over District alone, the site of countless revival camps, was responsible for the creation of also the Millerites, Seventh-Day Adventists, Jehovah's Witnesses, the Shakers, and more. Essentially, protestant denominations were popping up like daisies during this time, and specifically in the same place as Mormonism. I also want to point out that this time period of religious fervor and new religious thinking was not unique to the historical American religious experience. Many other Protestant denominations have begun in the United States including the Baptists, Methodists, Pentecostalism, Dandyism, and Evangelicalism (as much as one can define Evangelicalism in one basket instead of the fluid self-described religious "movement" that it is).
Anyway, back to Mormonism. Joseph Smith was unique in his day. All of these other denominations usually had some new interpretation of religious orthodoxy and orthopraxy. Seventh-Day Adventists believe Saturday is the sabbath, the Methodists believe they ought to try different methods of connecting to the holy spirit and they should discuss their experiences, the Pentecostals believe in being saved again and speak in tongues when the holy spirit embodies them... my point is that new denominations that pop up never rewrite the Bible, they just interpret it differently and practice differently. Not Joseph Smith.
Joseph Smith was a self-described prophet. The founders of other United-States based denominations were more of preachers with new ideas of how to be pious. No one claimed to be the likes of Abraham or Moses. Joseph Smith did. After reading the golden plates that God sent to him with His word on it, he wrote a new Bible (now, The Book of Latter-Day Saints) that practically all other Christians considered to be heresy. For many years after that Joseph Smith and his followers would migrate around America trying to find a permeant home, settling new towns and being ran out by the locals. Eventually, with the very brief introduction of polygamy essentially resulting in Joseph Smith's murder, Brigham Young would migrate the Mormons out to what is now Utah.
In short, no, there were no religious leaders, much less any prophets to compare to the likes of Joseph Smith at this time. There were important religious leaders, before, during, and after, indeed who would go on to found denominations and shape American religion for a long time, but no prophets, no one quite like Joseph Smith and the Mormons.
Sources:
Texts:
Gaustad, E. S., & Noll, M. A. (2003). A documentary history of religion in America: To 1877. Grand Rapids: Eerdmans.
Carroll, B. E. (2000). The Routledge historical atlas of religion in America. New York: Routledge.
Course: Religion in America (REL 220), Michigan State University, Professor Amy Derogatis.
TL;DR: 19th century America had its fair share of religious movements and new denominations being founded, but none compare to the infamous prophet of Joseph Smith. Because of his radical new interpretation of Christianity and imposition of unorthodox living conditions (forced migration, polygamy), no one compares in that time.