Yes, in the 19th century many states allowed non-citizens, or people who intended to become citizens, to vote in all types of elections. There is certainly no clear Constitutional ban on non-citizens voting either. If you had to make an argument, it would be that many of the suffrage amendments refer to "citizens." But, that's pretty weak, perhaps a state could allow non-citizens to vote, but would be able to have a poll-tax for non-citizens though.
http://en.wikipedia.org/wiki/Right_of_foreigners_to_vote_in_the_United_States