Lacking statutory limits or contractual agreements (including collective bargaining agreements) to the contrary, nothing will prevent the employer from requiring employees to consent to searches of their bags, cars, etc, while on company property, as a condition of continuing employment. And lacking statutory or contractual limits to the contrary, nothing prevents an employer from adding such a policy at any time they want to after someone has started employment.
Randomly searching vehicles is a good way to really kill employee moral. So I've never seen it happen. But it could happen, and especially if there is ever any cause to do so....including a fellow employer getting all nervous about whether the resident gun nut has scary guns in his car in the company parking lot in violation of company policy.
Utah's Parking Lot Preemption Law protects most employees from adverse employment action for having a gun (or religious items) in their private cars even if parked in the company parking lot. An employer can still require employees to allow a search of their car, and can terminate employees who refuse. But if they happen to find a gun (or Bible, or Jehovah Witness tracts, or a Quran, etc) in the car, they cannot take any employment action as a result. Of course, if they find company property that shouldn't be there....
But lacking a similar statute, employees either have to comply with no-guns policies even in parking lots, or they are relying on not getting caught and should consider the total of their conduct so as to minimize the chances of their policy violation coming to light.
Charles