Scott is correct, though the information is slightly outdated as of today.
The DOJ
decided late last year [pdf] that it was unconstitutional to apply a different standard to resident aliens than it was to citizens. A few weeks back, a
new 4473 [pdf] was sent to dealers, and the 90 days' proof-of-residency requirement no longer applies.
Resident aliens may own firearms. In fact, many states will issue carry permits to them as well. From a purchasing perspective, there are only two differences:
- A resident alien needs to provide his green card,
- the FBI NICS check will most likely return a "delay" result.
EDIT: Alland beat me to it. Dealers should be using the new forms as of today.