Can an immigrant vote

No, this is not permitted, although there have undoubtedly been a few occasions when someone did vote illegally and was not caught. Such occurrences are rare, however; non-partisan organizations that study voter fraud have found that it seldom happens. That said, in order to vote legally, you must be a citizen (and some states require more levels of proof than others); you must also be registered to vote, and thus on the official voter list.