They
can report it. The thing is, most simply don't report it because it's
not required. (The emergency approval requires it, but doesn't define how long after vaccination it is required)
My kid had an SVT 2 days after receiving 4 simultaneous vaccines. No prior issues, no family history, no cause determined by the doctors. They said that couldn't possibly be related, but when asked for any data/explanation on that theory they had none. (Later doctors said there could be some sort of connection) So they're telling me that they don't know what the cause is, but that it's definitely not the vaccines, even though there's no evidence to rule that out... yeah, ok. There's the possibility of coincidence, but it's astronomically small. The only way to tell one way or the other is to have the data from the larger population, which isn't happening because it's not being reported. I had to submit the report myself in that case.
I wanted to crunch the numbers myself, but the major issue is that even a small number of unreported cases will greatly skew the results when dealing with rare issues. You basically need anonymized medical records (I know, those don't really exist) so that you have a known population size and any medical treatment would have been recorded even if not reported.