The Halacha is that a Jewish woman may not marry a convert of Amon and Moav. Is there any way to know who is a descendant of Amon and Moav today?
No, the nations were mixed up in the days of Sennacherib the Assyrian.
When Sannecherib, King of Assyria, arose, he confused the identity of all the nations, mixing them together, and exiling them from their place. The Egyptians that live in the land of Egypt at present are of other nationalities. This also applies with regard to the Edomites in the field of Edom.
Since these four forbidden nations became intermingled with all the nations of the world [with] whom it is permitted [to marry once they convert], all [converts] are permitted. For when anyone of them separates himself [from them by] converting, we operate under the presumption that he became separate from the majority. Therefore in the present age, in all places, whenever a convert converts, whether he be an Edomite, an Egyptian, an Ammonite, a Moabite, a Kushite, or from any of the other nations, whether male or female, he or she is permitted to marry among the Jewish people immediately.
This Rambam is based off the gemorah in Berachos 28a
Interestingly, many poskim today don't allow DNA to have true halachic standing.