If the people are Jewish (by halacha, not “culturally” or “genetically”), and they are living together as man and wife, and its their child, the only possible objection to the marriage is that the woman didn’t go to mikva. It might be an issue if they they weren’t living together, or if the child wasn’t there child, or the man was married to another woman -but what the question describes is a proper Jewish marriage by a couple that is already de facto married, and wants to “make it official” for a legitimate reason. Consider if the reason was so that they could easily file taxes as a married couple or to own a house by the entireties (since it is very hard to get a marriage recognized without the expected paperwork, even though it is a valid marriage).
They most probably would need a “get” to marry anyone else, as well as a government divorce, since New York, like most states, recognizes the validity of religious marriage ceremonies even if you don’t have the marriage license (the couple and the clergy might owe a fine, but that’s a revenue measure, not a domestic relations matter).