In Belgium, "sans papier" is basically a "nicer" way to describe an illegal immigrant (i.e. a foreigner who has no valid permit(s) to live nor work in the country where he is residing), simple as that.
It doesn't sound nice but in the end it is what it is.
But for your translation purpose "undocumented immigrant" might be good.
Assuming the article is about people from elsewhere who currently live in Switzerland (or France, or wherever it refers to), not about Swiss (or French, etc) people who currently live elsewhere, I see no particular reason to change it from "immigrant" to "emigrant" or "migrant". You can if you want, of course, but I'd stick with the more commonly used term.
One can also use a similar expression when one has no documentation when travelling: voyager sans (ses) papiers or ne pas avoir ses papiers. This would be different context situations.
Some immigrants come in both categories, others only in the first. In the second category, it can be anybody who lost or forgot his ID before leaving home.
I am reluctant to tell people what to say in English because it is a foreign language to me, but I vote for the translation "illegal immigrant". it's the one avoiding the ambiguity explained above.
On this website, you can click your way through the languages, at least on some of the pages.
I had considered leaving it as 'sans-papiers' initially but the way we would use this in English just didn't fit with the context of the text I had to translate.