Illegal aliens are not immigrants ~ and even legal immigrants who are not citizens are not allowed to vote.
Whether or not illegals vote is irrelevant ~ the state may prohibit just all sorts of folks from voting ~ immigrants, illegal aliens, tourists, business travelers from abroad, folks in airplanes flying from one country to another, prisoners, chillun', etc.
I don't think the arguments against AZ's restrictions on illegals voting will get past even the most leftwingtard of justices.
Actually, there's nothing in the Constitution or its amendments which speaks at all about immigration. The Constitution does talk about naturalization - which is the method by which an immigrant becomes a citizen.