Short answer: because Mormons.

Historically, the Mormon church believed that some sins, including murder, could only be atoned for by the sinner being killed in a way that spilled his blood- so-called "blood atonement". For this reason, condemned murderers always had the option to be shot rather than hanged. In fact, for some time they also had the option of death by beheading, though nobody ever chose it.

Various pre-statehood Deseret News articles describe this. For example, on page 2 of the May 16, 1879 issue the execution of a murderer named Wallace Wilkerson by firing squad is described:

He has atoned for that deed as far as it is possible to do so by the pouring out of his own blood. Thus the divine law has been executed and human law honored. The culprit preferred shooting to hanging or decapitation. This was his privilege under our local statutes.

The article goes on to claim that shooting is more humane than hanging, as "there is something barbarous in the idea of strangling a man to death like a dog". This theme reappears in an article a few years later (May 10, 1882, page 8) in response to a New York Herald article about the hanging of the murderer William Sindram in New York City (who had apparently taken some time to die). The Herald mentions blood atonement as a reason why condemned criminals in Utah were shot not hanged, but repeats the long-standing accusation that the doctrine allows Mormon church leadership to order the assassination of those who had sinned against the church, in particular apostates- the Deseret News article is rebutting it.

The accusation did, however, persist (and to some extent still does). In 1889, the Church Presidency issued a declaration denying it, saying that the only punishment church courts could impose was expulsion from the church, and describing the killing of apostates as "abhorrent". An interesting quote from there, though, is

we regard the killing of a human being, except in conformity with the civil law, as a capital crime, which should be punished by shedding the blood of the criminal after a public trial before a legally constituted court of the land.

(My emphasis)

So at this point, the Mormon church still believed that judicial executions should shed the blood of the offender. This appears to have changed by 1966, when in the second edition of Mormon Doctrine, the Mormon leader Bruce McConkie removed an article from the 1958 first edition, in which he had stated that shooting would create blood atonement but hanging would not. In a letter explaining the change, he stated that "death is death" and that the reference to shedding of blood is a figurative expression which can refer to any legal execution. By 1978, McConkie issued a statement from the Church Presidency stating that "We do not believe that it is necessary for men in this day to shed their own blood to receive a remission of sins".