Utah's women were given the right to vote by the Utah Territorial Legislature, following the lead of their sister-state Wyoming. Due to timing of election dates women in Utah were the first in the nation to exercise this new power when Sereph Young votes on Feb 14, 1870.
The Church gave to its women the first exclusively women's organization in all the world; and it was representatives of this organization in mass-meeting assembled to enter their vigorous protest against the pending federal legislation which was intended to affect them seriously in their lives. Note that the Relief Society President used to be a life-long office. Not all Mormon women were members of the Relief Society; you had to be admitted by a vote.
Easterners concerned with breaking up the Mormon political control wrongly thought by giving women the right to vote they would throw off the tyrannical shackles of patriarchal polygamy and join with local nonmembers in removing Church influence in politics. Instead, the Church had correctly assessed that giving the women the right to vote, while their husbands were disenfranchised, would keep the church in control of the territory (as opposed to ceding control to the non-members in Utah.)
Utah women had the right to vote, but not the right to hold office. Female suffrage was ended in Utah by the Edmunds-Tucker Act of 1887.
[Tungate, Mel, Mormon Polygamy, http://www.tungate.com/polygamy.htm]