Technically, an "immigrant" who has been naturalized can vote. However, "illegal immigrants" have said that people can't be illegal, and they are not all called "immigrants". This letter writer was just being politically correct.
You are advised that if your residence in this country is illegal or you are an immigrant, voting in a federal election is a crime that could result in jail time.
As stated above a distinction is made via the conjunction or that states that if your residence in this country is illegal or if you are an immigrant. no modifiers are linked to the word immigrant, so this states that the writer infers that all immigrants cannot vote. Twisting the sentence around to get whatever meaning one hopes for is useless.