It may seem trivial to some -- "just vote somewhere with a shelter" but it's worth noting that homelessness as a state is an extreme burden and these kind of options simply may not exist depending on an individual.
Very poor people tend to have neither. Obviously they don't have money, but they also tend to be overwhelmed with the daily struggle to stay alive, or emotionally beaten down by dealing with various kinds of trauma.
That means even when homeless people manage to vote individually, they aren't likely to find politicians who cater to their needs directly. Instead the best they can hope for is a politician who caters to middle-class and wealthy people who care about the abstract concept of homelessness and have their own ideas about how to address it.
Why on earth is a home address needed to vote? Of course you need to vote where you are registered, I cannot vote in another city beside mine too, but even if I lose my house today I can keep voting in my city until I have a document that proves my identity, eventually for all my life.
Pardon my rant, it is not towards you or your comment, but some policies in the US are simply crazy.