Let me ask what may be a stupid question here.
When I ask "why don't these people have state issued IDs", a very common response is that people can't get to the facility where they are issued. It's not so much the cost ($20 for an ID that lasts 4-5 years in my state), or that they don't have the right paperwork (in some cases, anyway), it's getting to/from the facility and getting the time off from work.
So here's the stupid question. Why doesn't the state come to them? Invest in a few large mobile vehicles and drive to the areas (perhaps in the evenings) where there are a lot of voters without state IDs.
There might be a good answer to that question, but I can't think of one offhand...