Hello,

The maintenance module's responsible field will show only your internal users. If you want to display the other company's name, then you can create them as a user and select it in that responsible field. 

If you don't want to give access on your internal database, then you just create maintenance team with that company's name and select that team when you are creating the request. But please note that the company can't know what requests you created in this case.

You can add them as a contact in your database and follower of that request. Then use the Send Message feature as well to notify them.