Utah was definitely not a theocracy at the time suffrage was obtained.
It's actually a fascinating story. Mormons largely supported suffrage, but had to fight the federal government tooth and nail to get it.
In 1869, Utah was under territorial rule, and the federally-appointed government was determined to limit the influence of the LDS Church and end polygamy. The legislature even passed female suffrage during their efforts to rid polygamy from the state.
But it had exactly the opposite effect; most woman voted in favor of allowing polygamy. Thereafter, some national supporters of women's rights actually became opposed to women's suffrage in Utah because of it was being used to support polygamy. (Enfranchisement all fine and well until someone disagrees with you.)
In 1887, Congress redoubled its efforts to end polygamy, and revoked the ability of woman to vote that the legislature had granted earlier.
The LDS Church supported the creation of the Utah Women's Suffrage Organization, to try and restore the right to vote. A Utah Territory judge sided with the women's movement, but the Supreme Court rejected it, citing Congress' law.
Not until the Church renounced polygamy, Utah obtained statehood, and the women's right to vote was included in the state constitution, were Utahns able to undo the Congress' targeted repeal of suffrage.
Utah women then helped secure the Nineteenth Amendment[1].
[1] https://www.utahwomenshistory.org/wp-content/uploads/2018/02... (The man in the picture was Senator Reed Smoot, who was also an Apostle in the LDS Church.)
