I’m guessing the plates in the cleaning co. name, not yours. A lease agreement onboard wouldn’t hurt. Just in case of an inspection or accident. Probably never need it. If the plates in your name, no need. Register it with IFTA in Trucing Co.s account. No different than someone leasing onto someone else’s authority and bringing their own plate. I may be wrong, but makes sense to me.
The other question is if your box truck don't go out of state you will need only a DOT number not an MC number with it. On the other hand if you do go out of state with box truck then you can lease the truck unto the company with the MC number https://www.csatag.com/files/128858262.pdf
That same site has a copy of a standard lease form