I don’t have a specific answer, but keep in mind that the astronauts were always on the moon in sunlight. The reflected sunlight would tend to wash out the stars (cue no-moon-landing conspiracy theorists), so a picture of the Milky Way would probably not turn out too well. Also, they were rather preoccupied with the moon, since, that’s why they were there.
Long exposure times require a tripod or some other means of stabilizing the camera. In a situation where every ounce counts, I doubt that they prioritized bringing one along just so that they could take pictures of something that can be photographed from Earth’s surface or from orbit. The moon itself was brightly sunlit, needing only short exposure times (and was the subject that they were interested in photographing.)
Plus, they forgot to paint a copy of the Milky Way on the studio backdrop.
If I recall correctly, all six landing sites were on the side of the moon facing Earth. The light from the planet would make it difficult to take a picture of the surrounding stars.
I’ve only seen one picture with a starfield, and that includes Earth (overexposed). Even that one was, I think, in completely the wrong direction to see the Milky Way.
Even if they had been in the right direction, I’m not sure that the camera technology was up to snuff in the 60’s. They could do long exposure times, but even today it takes a reasonably high-end camera set up in just the right way.
This is the camera used by Apollo 11 astronauts. It appears this camera had a mechanical shutter with a fixed shutter speed of 1/250 s. Aperture dial goes from F/2.8 to F/22; the instructions say to use 5.6 in shadow and 11 in sun. Anyway, there’s no way you can capture any star with such short exposure time.
If it weren’t for the fixed shutter speed, this camera would be perfectly capable of capturing the milky way, if it is carefully shielded from sunlight and stray light from the surroundings and fixed on a tripod (which the Apollo astronauts didn’t have either, as far as I know).
I’m not even sure why they’d have bothered to do so. The picture would look absolutely identical to one taken from earth’s surface. The Moon’s orbit around the earth is about 0.25% of the Earth’s around the sun. So even if you happened to have pictures from the moon six months apart and at full moon time, you’d only get that percentage change in your parallax measurements.
Light absorption by the atmosphere is not big enough to be noticeable. Distortion by the atmosphere is only evident at high magnification (i.e. through a telescope). So no, it wouldn’t make an appreciably better photo of the milky way.
Light pollution is the issue.
Even in dark-sky locations, there is significant light pollution, and even if there were no artificial light sources, the atmosphere itself is a significant source of light: