Keep the container wholly on private property, on your own driveway or parking pad and clear of the sidewalk and the alley, and the city asks nothing of you.
Street placement is the awkward part, because Calumet City publishes no route for it. Its streets chapter carries a driveway permit, an excavation permit and rules for oversize vehicles, but nothing covering a container in the roadway, and no fee is posted for one. The solid waste chapter does allow a compliant container to be set out for collection without that counting as an unlawful street deposit, which is a narrow saving rather than a permit. So plan a placement on your own ground, and if there is genuinely no room, speak to the city first, because an administrative route may exist that is not written down. We will not promise a roadway placement here.
The licensing question falls on the hauler and it is a real one. Calumet City makes it unlawful to engage in the business of scavenger, or the collection or disposal of refuse of any kind, without first holding a city license. The application goes to the city clerk and the clerk sends it to the city council, and no license is granted except on council approval. It runs out at the end of the calendar year whoever holds it, the fee is set per truck above a base figure, and the city requires public liability cover plus workers compensation, with the license becoming void if that cover lapses without the city's consent.
That is a licensing city rather than a closed one: there is a published way in, no approved-hauler roster exists and no annual enrolment window applies. Your account manager confirms the position before a delivery date is set. The city's department directory will put you through to Public Works or the clerk's office, and City Hall is at 204 Pulaski Road.
One further point if the container is going to a rental property. Where an owner responsible for paying for scavenger service refuses to pay and the service stops, the city can step in, pay and recover, and the unpaid amount becomes a lien against the property. That makes the billing arrangement worth getting right at the start.