The Utah Fit Premises would be your starting point, which states: "for buildings containing more than two residential rental units, [the landlord must] provide and maintain appropriate receptacles for garbage and other waste and arrange for its removal, except to the extent that the renter and owner otherwise agree."
The language of the statute does not specify that trash removal must be done by the city, but it does require that the receptacles must be appropriate (a truck bed probably doesn't count) and that arrangements for removal must be made (I think it can be implied that removal must be reasonable in timing).
The Act also states that the landlord must "maintain common areas of the residential rental unit in a sanitary and safe condition." The conditions of the trash area sound like they violate this provision.
In order to use the Fit Premises Act, you have to provide written notice to the landlord. I've attached a link to the Utah Legal Services guide which will help you get started.