I think he may be referring to this chapter of the Chandogya Upanishad, where the king Pravahana Jaibili tells the sage Uddalaka Aruni about the journey of a virtuous soul if it is reincarnating:
But they who living in a village practise (a life of) sacrifices, works of public utility, and alms, they go to the smoke, from smoke to night, from night to the dark half of the moon, from the dark half of the moon to the six months when the sun goes to the south. But they do not reach the year. From the months they go to the world of the fathers, from the world of the fathers to the ether, from the ether to the moon. That is Soma, the king. Here they are loved (eaten) by the Devas, yes, the Devas love (eat) them
Having dwelt there, till their (good) works are consumed, they return again that way as they came, to the ether, from the ether to the air. Then the sacrificer, having become air, becomes smoke, having become smoke, he becomes mist, having become mist, he becomes a cloud, having become a cloud, he rains down. Then he is born as rice and corn, herbs and trees, sesamum. and beans. From thence the escape is beset with most difficulties. For whoever the persons may be that eat the food, and beget offspring, he henceforth becomes like unto them. Those whose conduct has been good, will quickly attain some good birth, the birth of a Brâhmana, or a Kshatriya, or a Vaisya. But those whose conduct has been evil, will quickly attain an evil birth, the birth of a dog, or a hog, or a Kandâla.
The story of Pravahana and Uddalaka Aruni is originally from this chapter of the Brihadaranyaka Upanishad, and Pravahana says the same thing there too:
But they who conquer the worlds (future states) by means of sacrifice, charity, and austerity, go to smoke, from smoke to night, from night to the decreasing half of the moon, from the decreasing half of the moon to the six months when the sun goes to the south, from these months to the world of the fathers, from the world of the fathers to the moon. Having reached the moon, they become food, and then the Devas feed on them there, as sacrificers feed on Soma, as it increases and decreases 1. But when this (the result of their good works on earth) ceases, they return again to that ether, from ether to the air, from the air to rain, from rain to the earth. And when they have reached the earth, they become food, they are offered again in the altar-fire, which is man, and thence are born in the fire of woman. Thus they rise up towards the worlds, and go the same round as before.
So insofar as the defining characteristic of life is the fact that it is animated by an atma, and insofar as the souls of virtuous individuals enjoy themselves in the Moon before they come down to the Earth as rain, it can be said that living things "come from the Moon". This isn't the same as saying that the biological material (DNA and cells) that living things are made of come from the moon, but I think it's what Vivekananda was getting at.
The one thing that militates against this interpretation of Vivekananda's words, however, is that it's not true that all souls go through the Moon before coming to Earth; Adhyaya 3, Pada 1, Sutra 13 of the Brahma Sutras says that that the souls of unholy individuals do not go to the moon. Here is what Adi Shankaracharya says about this Sutra:
It is not true that all men go to the moon. For the ascent to the moon is for the purpose of enjoyment only; it is neither without a special purpose nor for the mere purpose of subsequent re-descent. Just as a man climbs on a tree for the purpose of breaking fruit or blossoms, not either without any aim or for the mere purpose of coming down again. Now it has been admitted already that for those who do not offer sacrifices there is not any enjoyment in the moon; hence those only who perform sacrifices rise to the moon, not any other persons. The latter descend to Samyamana, the abode of Yama, suffer there the torments of Yama corresponding to their evil deeds, and then again re-ascend to this world. Such is their ascent and descent; as we maintain on the ground of such a course being declared by scripture.