Yes, there are people who can’t obtain an ID card, for whatever reason. A European citizen who couldn’t obtain an ID card would have the exact same problems voting that an American citizen does. I don’t have a systemic solution for that. This would seem to be something that would need to be handled on a case-by-case basis, possibly involving the judicial system and a court order. It also doesn’t seem to be a particularly common problem. I’d bet all the money in my pockets that OP does, indeed, have some sort of ID card.

We have a remedy for this: Provisional ballots. Cast your vote now, and resolve any clusterfuck with registration later.