On private property, like your own driveway or a fenced job site, no placement permit applies in Melbourne. What does apply is a separate city requirement tied to the work itself.
Melbourne's solid waste rules give the City's contracted hauler the collection rights, then carve out firms doing building or remodeling work that needs a city building permit. That carve-out runs through a trash hauling permit issued by the city manager's office, and it is the instrument that makes a private container lawful on a permitted job. The City sets the fee as a percentage of what your hauler bills you plus a per-cubic-yard sanitation charge, and starting the work before the permit issues can double the fee and add a separate penalty. In short, the permit belongs in the schedule next to the building permit, not after delivery.
Street and sidewalk placement is a firmer answer. The City's streets rules make it unlawful to obstruct any part of a public street, sidewalk, parkway, park or plaza with building material or any other obstruction, and the chapter offers no permit to lift that. Plan on a driveway or on-site placement in Melbourne.
The City's own construction site rule runs the other way and helps you. A construction or demolition contractor has to provide bulk or detachable containers sufficient for the loose debris, paper, building material waste and scrap the crew produces, with the number of containers and the service frequency set by how much the job makes. Anything scattered onto neighboring property has to be cleared by the end of that work day.
If you are not sure whether your job needs the trash hauling permit, your account manager will get you the current number for the city manager's office and walk the application through with you before you book.