Theocracies can be republics (basically, any non-monarchic system is a republic), and can have elected representatives.

The Islamic Republic of Iran, for instance, meets both.

Now, the assertion that Utah is effectively a theocracy rests on the claim that the leadership of the Church of Jesus Christ of Latter-day Saints asserts informal power loosely analogous to that held formally by, say, the Guardian Council in Iran. I’m nit saying that that’s true. But that it is formally a republic with democratic election of representatives isn’t even slightly incompatible with it being an informal theocracy, you can even be a formal theocracy with those features.