How do I apply for a visa for my Japanese girlfriend, are we able to get married in the USA if she doesn't have a visa?

A foreign national may marry a U.S. citizen in the U.S. and then complete the Adjustment of Status application process to become a Lawful Permanent Resident (to get a "Green Card"). Beware, however, that details about your fiancee's entry into the U.S. may be problematic. Some nonimmigrant visas, such as "visitor's visas" are inappropriate for someone who is engaged to be married to a U.S. citizen and who intends to permanently reside in the U.S. - visitor's visas are exclusively for people who genuinely intend to stay temporarily and then return abroad, and misuse of a visa can constitute "visa fraud" with very harsh consequences. Other nonimmigrant visas, such as H1B visas, are suitable for someone having both an intention to stay temporarily and an intention to stay permanently, if authorized. It would be wise to consult with an immigration lawyer BEFORE your fiance enters the U.S. in order to discuss relevant details about your fiance, your marriage plans, etc. so that the attorney can advise about eligibilities, options and strategies. Some immigration law firms, including mine, offer legal services on a "flat fee" basis so that a client will know the total expense from the very beginning, and a few immigration law firms, including mine, offer an initial consultation free of charge.