For the record, I'm a resident alien, living in Washington state. Washington state law requires resident foreign nationals to acquire a state Alien Firearms License before they are allowed to "possess, carry, or have control over a firearm." This is very much a "shall-issue" type thing; the primary requirement is that the applicant produce a letter from his consulate stating that he does not have a criminal record in his country of origin. This requirement is waived if the applicant can demonstrate that his consulate refused or failed to issue such a letter within 90 days of his requesting it, or if the country in question has no diplomatic representation in the state. Essentially, it's a background check, and I for one have no problem with it, any more than I have a problem with American citizens with a felony conviction being prohibited from owning firearms.
New York state, by contrast, prohibits all non-US citizens from possessing any "deadly or dangerous weapon." This, I have bit more of a problem with, though it's not so much about the weapons as such, but rather, the constitutional ramifications. Essentially, New York state claims that legally resident aliens do not have the right to keep and bear arms; that implies that legal aliens are not part of "the people" (you know, whose right "to keep and bear arms shall not be infringed," and while you can quibble over what really constitutes infringement, there's no question that a blanket prohibition is infringing). That in turn implies that much of the Bill of Rights supposedly doesn't apply to legal aliens: the second half of the First Amendment, the first half of the Fourth, all of the Ninth and Tenth speak of rights reserved to "the people."
Okay, the Fourteenth Amendment guarantees all persons on American soil due process and "the equal protection of the laws," so you're (notionally) protected from arbitrary search and seizure, but I'd still like to be able to assemble peaceably and petition the government for redress of grievances.