A container on private property, in a driveway or inside a fenced job site, is the normal arrangement in Thomasville and no city permit attaches to it.
On the licensing question the answer is unusually clear, and it is worth stating plainly. The city's solid waste chapter creates only two offences: putting waste in a container that is not yours without the owner's consent, and interfering with city collection crews or their equipment. There is no collector license, no hauler registration, no franchise and no exclusivity clause anywhere in the chapter. The city's business chapter licenses a long list of trades and names no waste activity at all. So nobody needs the city's permission to bring you a container here.
The city goes further and puts the duty on your side of the line in terms. Construction, demolition, land clearing and landscaping waste is defined as its own category, covering discarded construction materials, lumber, masonry, fixtures, stumps, trees, brush and dirt, and the city's rule is that all of it must be collected, removed and disposed of by the contractor or builder, or failing them by the occupant of the property. Commercial and industrial waste is treated the same way, as the job of whoever creates it. A city that writes that down is a city expecting a container on site.
The street is the one place to take care. Thomasville's streets chapter has no container permit of any kind. What it does have is a rule making it unlawful to place building materials on a street or sidewalk so as to obstruct it, and a second rule reaching boxes, casks, barrels, crates and any other like obstruction, where every day the obstruction stays counts as a separate offence once you have been told to move it. The city manager controls all work done on the streets and sidewalks. So a public-way placement in Thomasville is a conversation with the city rather than an application, and your account manager will have it for you before the truck is scheduled.