By marriage and living in the US your wife is an immigrant alien, whether she has a green card or not. It is only non-immigrant aliens that cannot own firearms, and even they can under certain circumstances.
Not necessarily true. To bring your spouse into the US as an immigrant, you must file a petition with the embassy.
While that's in progress, she can apply for a non-immigrant visa. But since she's a Japanese citizen, she can enter the US as a non-immigrant anyway, under the Visa Waiver Program
. Either way, as a non-immigrant alien, she can be in possession of firearms and ammunition if
she has a valid hunting license issued from a US state. (There are other possible ways for non-immigrants to posses firearms but they are far-fetched, such as having a waiver from the US Attorney General for example).
Bear in mind that "possession" can be just living in a house where firearms are kept, or being in the same vehicle, so you will want to make sure your wife can legally posses any firearms you intend to buy.