Under Utah law, a DUI conviction within the past 5 years makes you ineligible for a Concealed Firearm Permit (CFP).
Utah Code §53-5-704(2)(b) lists disqualifiers, including:
Other bars: Felony, domestic violence, protective order, adjudicated mental incompetence, unlawful drug use, dishonorable discharge, renounced citizenship.
If your DUI is >5 years old and it's your only one, you may apply. You'll need to disclose it; the Bureau of Criminal Identification (BCI) will review. Consult a Utah firearms attorney if uncertain.