Arduino Cellular Shield EMC Testing for FCC
This is a great article for you on the subject based on  Part 15 of Title 47 CFR. I do not have a huge experience on this but when I have dealt with parts that are part of an assembly, such sub assembly was not tested. Why? for example the same board would be use in multiple chassis each one having a different EMI signature and shielding. Also differences in power and clocks.

Given that it is an intended radiator, it rises flags. The end product must have the correct certifications. In order to do that, the owner of such product has a better time using something that has pass certifications in someway, since they do not have control of the product. The main aspect of the certification is that you limit the user to the antenna type and modulations used.

You can ship modules without certification all day long, as long as who is receiving it does not power it up ;) but you should at very least make sure you are not setting someone up for failure.

Shipping to other countries in big quantities have it's own rules as well. Things like RoHS and REACH. Since it is a sub assembly I have not seen many issues unless it carries installed batteries or it has protected IC per ITAR etc.

Again not an expert but some experience of what I have seen.