A container on private property, your driveway or a fenced job site, needs no city permit in Lemoore, and the city's code is unusually helpful about construction debris generally.
Lemoore makes its own refuse collection service mandatory for every premises in the city, and it is unlawful for anyone other than the city to haul away food waste or to take recyclables out of city recycling containers. But the city's code then says in terms that material which is not food waste may be collected and disposed of by private haulers, who simply need a city business license.
Better still, the city's definition of the general waste category expressly excludes any material left over from the repair or construction of buildings, sidewalks, or other improvements. Construction debris is not food waste, it is not on the city's closed list of household material, and it is carved out of the general category by name. On the face of the code it sits outside the city's refuse system altogether.
The vehicle requirement is a performance standard rather than an equipment specification: a vehicle carrying this material must have equipment to stop the contents blowing out, falling, or otherwise escaping. A secured cover meets that on its face, which is not true in every California city.
The county layer does not reach you either. Kings County issues a non-exclusive hauler license that the county must grant where an application is made, and it covers the unincorporated area only, so it does not apply to a job inside the Lemoore city limits.
Public Works is at 711 West Cinnamon Drive and can be reached on 559-924-6744. Your account manager confirms the licensing position and the business license question before delivery, and will raise anything about a street placement with the city for you.