Question

how did nasa discover that there is no oxygen in moon

Answer 1

There is no sharp difference between "no air" and "very little air". That it is "very little" is seen when we observe the Moon with a telescope. An atmosphere creates a visible haze, especially on the edge of the object (because of refraction). Especially well visible it would be during Solar eclipses. So when people started to observe the Moon with telescopes it was clear that there is very little air. The more powerful the telescopes that were used, the smaller this "very little" became.

For practical purposes it was clear that there is no air by the end of the 18th century.

For comparison, comets usually have atmospheres (visible with a naked eye) but they are so rarified that "there is no air" on the comets, for all practical purposes.

EDIT. Of course when I say that "it was known in 18-th century" this means "it was known to the experts, scientists". It is easy to show that this was NOT known to the "general public". Many writers (and some movies in the beginning of 20-th centiry) described travels to the Moon, and they did not know or did not care that a human cannot breath there. Jules Verne's travelers, for example has no provision for breathing on the Moon. Jules Verne wrote in the second half of 19-s century. (That air is necessary for survival was shown in 17-s century).