Backup codes could work - but if they have the support of a trusted person they likely can be assisted in other ways, too.
Defining a state-sponsored email account that can only be logged in from specific government machines (imagine a kiosk at the DMV, say) where there are trained clerks who can identify homeless in some way could work.
An interesting idea, but I suspect it just pushes the issue back one more step. How do you authenticate for login to that email account? Specific machines limits but doesn't fundamentally change the attack surface.
If the person has ID, then many options work, but if they don't what can a DMV and trained clerks do that others can't in some way?
Lastly, I'm not from the US but even I've heard that the DMV is a hellish place with queues hours long. Putting more barriers in front of those who are already in a tough spot (and may need to spend that time working, queueing for shelters, etc) is a big ask.
Yeah, you have to keep falling back - my idea was that assuming homeless don't move very rapidly from one location to another, you'd have people at the DMV or shelter or wherever you put this who actually recognize the person and can "vouch" for them.
It's not an easy problem to solve with "one quick trick" by any means. Part of the reason the DMV can be hellish (in the US at least) is they have to deal with everyone who has an ID, not just the "good customers".
Defining a state-sponsored email account that can only be logged in from specific government machines (imagine a kiosk at the DMV, say) where there are trained clerks who can identify homeless in some way could work.