Groups of Mormons still survive in the wasteland, mostly in the area that used to be known as the state of Utah, which originated as a safe haven for the Mormons as they tried to escape persecution by establishing a community, which became Salt Lake City. This became the state of Utah when the United States of America won a war against Mexico. Though truly brutal groups like Caesar's Legion will not hesitate to enslave or kill Mormons, most tribals and other organizations leave the Mormons alone, knowing that they often will voluntarily give medical or other aid to groups who need it. The people tolerate the Mormons' preaching because finding help with relatively benign conditions is rare.
In 2062, many Mormon congregations came together to purchase spots in Vault 70, located in Salt Lake City, Utah. As part of the Vault experiment, Vault 70 was assigned a clandestine social experiment―specifically, for the jumpsuit extruders to fail within six months of the vault's sealing. The eventual total lack of clothing combined with Mormon religious sensibilities resulted in the single largest block of social data collected during the Vault program.
The following is based on Van Buren and has not been confirmed by canon sources.
In 2190, 113 years after the Great War, Vault 70 opened and its residents used the three G.E.C.K.'s within to finally realize Joseph's Smith dream of a New Jerusalem, atop the ruins of Salt Lake City. Between 2220 and 2233, New Jerusalem's intolerant prophet and apostles repeatedly voted to have no commerce with outsiders from other, failed communities, refugees, or tribals. Finally, in 2233, many disgruntled and desperate refugees stormed New Jerusalem's gates and overwhelmed the militia; certain the Mormons were hoarding food and water while everyone outside the city walls suffered and died. Most of the Mormons were slaughtered. The survivors scattered into the desert.
Two years after the fall of New Jerusalem, the new living prophet, Judah Black, led most of the remnants of the Mormon community north to Ogden. There, they established the town of New Canaan. A year later, they and a group of squatters got the Jericho Water Plant running fresh water into the city. Judah Black died of old age in 2245, and two years after that, Jeremiah Rigdon emerged from a strange and powerful fever, claiming that an angel appeared to him in a vision, calling him to be the living prophet of God.
In 2246, the Mormon missionary Joshua Graham encountered two Followers of the Apocalypse, Bill Calhoun and Edward Sallow. The three eventually went on to become the founders of Caesar's Legion, bringing great shame to the Mormons. After Graham survived his execution after his failure at the First Battle of Hoover Dam, the Church accepted him back into the flock, incurring the wrath of Sallow, now known as Caesar. In 2281, New Canaan was burned to the ground by the White Legs, a group of tribals who were tasked with the destruction of all of Graham's people as part of their petition to be absorbed into the Legion, killing Bishop Mordecai in the process. 30 of the survivors managed to find each other in the chaos, and traveled to Zion Canyon under the leadership of Daniel. There, they encountered 4 tribes - the Crazy Horns, the Dead Horses, the Sorrows and the Tar Walkers. By the time the Courier arrives in Zion, only the Dead Horses and the Sorrows remain, the other two having fallen already to the White Legs. Daniel and Graham see it as their responsibility to ensure the survival of the native tribes - by any means necessary.
Behind the scenes
I think Mormons are interesting because they occupy such a unique position in American society. Since their early days, they've had a lot of conflicts with the people around them and rapidly pushed west, out of the Midwest, and eventually into what would become Utah. Events like Missouri Executive Order 44, Haun's Mill, and the Mountain Meadows Massacre show how violent that conflict could be at times. The society that they built in the Utah region was done with local tribes like the Paiute but apart from outside, mostly-European influences.
As a result of these conflicts and their eventual concentrated build-up of Utah, Mormons have been, and often still are, considered "other" by many Americans. Unsurprisingly, Mormon communities can be extremely organized and powerful. Unlike many other powerful religious groups, the geographic concentration of Mormons is quite dense, so I think it produces an interesting dynamic in American politics and culture. The military history of the Mormons (fighting against and for the federal government) and the central role of J.M. Browning in the development of many of the U.S. military's most notable weapons (the BAR, M1911 Pistol, and M2) throws another element into the mix.