The biggest issue is probably that earthquakes, at least on earth, are short-term events. The longest lasting ones are for mere seconds, while the longest recorded one was around 10 minutes. That does not include the quakes tapering off at the end or aftershocks. There is a lot of information on earthquakes, obviously.
From what I can find, due to the dramatic differences between the earth and the moon (molten vs. solid core, etc.,) "moonquakes" last quite a bit longer; large, shallow ones can hit a 5.5 on the Richter scale, and 10 minutes duration at maximum movement has been seen. Then they taper off slowly, up to being over several hours. There is relatively little data and a lot of unknowns on them.
So even though they aren't quite the same thing, these are still relatively brief events compared to say a high altitude satellite orbital period. Unfortunately, most satellites are always "moving" in their orbits. Well, I guess there must be a luna-synchronous (?)(!) orbit, but again, you would be locked into a limited observational area.
For a probe/satellite, even in a very high, slow orbit, it might only be able to face one third of the surface of the moon at most; at lower orbits this would be a smaller fraction of the surface. You would only be able to observe vertical movement; quakes are multidimensional.
Plus the peripheral observable surface would be at a very low angle instead of from above, making detection of any motion more difficult; the distance from the probe would not change in the case of vertical motion on the surface anywhere near the edges, and the angle would make both vertical and horizontal measurements less precise. In addition, the reflectivity of the surface at such angles would probably be far less.
So what we end up with is only periodic observation of the desired areas, yet we're looking for brief, finite events. The probability of taking a measurement coinciding with the quake event would be fairly low with a single satellite, at least for localized events.
This could be solved, however by using more than one satellite. A minimum of three in a very high equatorial orbit could start to approach full coverage at the lower latitudes, but practically at least twice that many would be required. Then you would also want to use different orbits for additional probes to observe the polar regions for which there is currently no data at all.
Also, data from the "dark side" wouldn't be in real time without some data relaying being provided to earth. Although I believe that a system may be in place for that, or in the works, not quite sure.
The four instruments that the Apollo missions placed on the moon worked for between 5 and 8 years each, providing a limited quantity of data already. But of course, that is only for the four specific locations that existed, whereas a satellite could observe many, many points in very large areas. But only some of the time.
The big advantage of being on the ground is that you can observe all motion directions at that location; and with the moon being tidally locked to the earth, direct communication is possible on the side facing the earth. You would still need data relaying on the other side.
The concept and use of technology is good, but perhaps the implementation would be more difficult than using localized measurement.
Comparative information on earthquakes.
NASA lunar seismometer and moonquake information.
P.S. I'm a total newbie here, so any feedback on whether this Answer is appropriate, acceptable or proper would be appreciated.