How do I find government cleaning contracts, and which buyers actually post janitorial bids?

Smaller janitorial bids are posted by federal, state, and local entities, but you will need to search different portals for each level of government. For federal contracts, you are correct to start with SAM.gov, but also look for opportunities below the simplified acquisition threshold (SAT) which may not be synopsized as broadly.

Federal agencies can use simplified acquisition procedures for purchases not exceeding the SAT (FAR 13.003(a)). Many smaller requirements, like those for a crew of two or three, fall into this category. These may be solicited directly from vendors known to the contracting officer or through systems like GSA eBuy for GSA Schedule holders. You can also contact federal agencies directly to learn about their upcoming needs.

For state and local government cleaning contracts, you will not find them on SAM.gov. Each state, county, and school district maintains its own vendor registration system and procurement portal. For Jacksonville, FL, you would need to register with the State of Florida's procurement portal and potentially separate portals for Duval County, the City of Jacksonville, and the local school districts. These portals usually have search functions for bids by keyword or NAICS code. An APEX Accelerator in your area can help you navigate these different state and local systems.