Must FCC-certified modules be labelled with the correct FCC ID?
Here's my understanding.

Since it is a radio module, then it is definitely an intentional radiator and if they claim it has been FCC-certified it should have an FCC ID on each module.  You should contact the manufacturer and ask why it's not there.



If you incorporate the module into your own product, and the module already has an antenna (e.g. a "chip" antenna like the one in the picture) then you don't have to do anything more except provide the same FCC ID on the outside of your product, or (as part of a recent rule change) in an accessible menu if your product has a screen.  In any case, the FCC ID must also be included in a label on the outside of the package.

However, if the module has no antenna, and you connect it to one you made (e.g. PCB trace antenna) or bought, then guess what, you have to do the FCC testing all over again(!) which can many thousands of dollars.

It's been a few years since I was directly involved in a product like that (cell phone modem + PCB antenna), so I can't remember if you are supposed to include both the FCC ID for the modem and your own FCC ID, or just the latter.  I suspect it's both.