As far as UDOT is concerned, any tire that is "manufacture rated" as a snow tire passes muster. Obviously the subjective judgement of tire acceptability hinges on the person camped on the side of the road. There is no official chart for every tire and/or vehicle.
From a personal experience point of view, any vehicle well suited to make the drive is considered acceptable. Any vehicle that appears to pose an obstacle, safety or otherwise, to other drivers will be given extra scrutiny. For all of UDOTs faults, common sense in this area doesn't seem to be one of them. They do a pretty good job at keeping the roads open.