Put the container on private property, a driveway or a construction pad, and there is nothing to apply for in Columbus. That is the normal arrangement, and it is also the cleanest, because the city's container rules are written around the city's own carts rather than around a rented container.
That distinction is worth understanding. Columbus declares a short list of nuisances about containers: setting one at the curb more than twenty-four hours before collection, keeping one open when the lid should be on, putting trash out in cardboard boxes or paper sacks, and letting waste turn odorous. Every one of those is drafted to apply to a city-owned or city-approved container, meaning a cart the owner or occupant of a residential property bought from the city. A roll-off you have rented is neither, so those rules do not reach it. Fines for a resident who breaches them start at fifty dollars and climb in fifty-dollar steps to a twelve-month cap of one thousand dollars.
On the collection side Columbus is genuinely open. The board of public works and safety is authorized to provide for the collection and disposal of solid waste from the premises of all residences in the city and from the various units of government operating in the city, and it may contract for that work. The grant stops there. It does not reach a commercial generator or a construction site, it creates no exclusive arrangement, and the city defines residential premises as single-family dwellings and multiple-family dwellings up to and including four separate living units per parcel. Neither the business regulations title nor the health and safety title carries a collector license, permit, franchise or registration of any kind.
Street placement is the one open question. Columbus has no obstruction chapter naming containers, so nothing specific is published about a container in the roadway, and we make no promise about one. There is a practical caution: Second Street and Third Street are state-controlled highways, and the city will not approve activity in them without the state transportation department first. City Hall is at 123 Washington Street, and your account manager can point you to the right office.