There are some times where I cannot select a particular wainwrite’s wagon, even after letting him move away from other objects. I go to others and can select them to see what they are hauling. I think the selection of the wainwright himself, either by clicking on him or from the building UI, should either show you what the wagon holds, or give you a button to click on that selects the wagon UI similar to how a building may have a button to select the construction site when it is being upgraded.
Evidence: In this shot, you see the cursor does not change to a spyglass while hovering over the ox. I had to use Windows Steps recorder to get this shot with the cursor as there is no in-game screen shot function that includes the cursor.