The U.S. Supreme Court paved the way for same-sex couples to obtain immigration benefits when the Court struck down the Defense of Marriage Act in 2013. DOMA had prohibited federal agencies including USCIS from extending marriage benefits to homosexual couples.
With the Supreme Court’s decision, USCIS began processing same-sex marriage-based green card applications. U.S. citizen spouses are now eligible to sponsor their spouses for lawful permanent resident status. This extension of immigration benefits even extends to states that do not recognize same-sex marriages. If the couple goes to a state where such marriages are legal (like Iowa and Illinois), their marriage will be recognized even if their home state does not recognize it (including Missouri).
This video explains the green card process for same-sex couples.