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.
It is, indeed, but the proper solution here is to lift them up to the bar, not lower the bar down to them.
Lack of ID prevents you from getting and keeping a job, attending school, accessing the banking system, getting a PO box, getting licenses. Being unable to vote is the least of your problems.
The proper solution is not to figure out how to make voting accessible to those without an ID. The proper solution is to get them an ID.