The joke is that instead of her being a wife (yome), he could be the husband (muko), which is a reference to the Japanese practice of families with daughters but no sons legally adopting a man and having him marry one of the daughters. The man would change his last name to that of his wife's family. The term for this is "mukoiri".
Basically he's saying that if she can't be a bride (with the standard implication of her becoming a part of her husband's family), a guy could just "be a groom" and marry in to her family, with the joke being the complementary meanings of "yome" and "muko".