The Vital Records office is not going to be a helpful Utah background check resource unless you are a close family member. All birth, death and marriage records are confidential for 100 years (except for death certificates, where it is 50). They'll release records to relatives as long as you can provide documentation of your family status.
Getting Criminal Histories
A more common Utah background check resource is the Department of Public Safety, where you can access someone's criminal history records. This is not open to the public though and is only a service for employers who are doing screening for hiring. Only certain employer classifications have access (such as those working with children) so others may have to request the individual to make their own request for a history file. The cost for your own history is $15 and they only serve records by mail or in person.