When nothing else is said, it is implied the owner can do as they please. It is also implied that you only do company specific things with the company equipment.
Whether that is a computer, that you do not use for private stuff or a power drill that you do not take home because you got a new TV you need to fasten to the wall.
With nothing else said, it is implied that the company has total control over company property.
The only limits frequently drawn by law are privacy concerns where a necessity of either living or the specific job is concerned. For example, you have to use the restroom multiple times a day, the company cannot have cameras there, just because they own the building. If you need to undress/dress for your work, the company cannot have cameras there, even if they own the building, the cabinets and even the work outfits. Because it's your privacy within the necessary restraints. It's not your free decision to use the restroom, you have no choice if you work a normal day.
Saving a private file on a work computer is not necessary and totally voluntary. If you do that, that is your own problem. Matter of fact, it is against the implied policy of using work equipment for work only.
There might be different agreements for bring your own device, company cars that can be used privately etc, but those will be explicit.
So if you cannot find an explicit exception, it is very likely the company has every right to search any of their own property, whether it's rooms, lockers or computers. There is no such thing as "your computer" or "your locker" in a company. It's their computer and their locker, that they allowed you to use for company business only.