Although there's no direct evidence to support this, there is some other "nonsense" from TOS which, in this light, suddenly makes sense.^ That is not what I was talking about.
Think of it this way: In "Mudd's Women", Harry expressly communicated ahead (surreptitiously, to be sure) to the Rigel XII mining colony using "subspace frequency 3-9". So we can assume that Starfleet personal communicators in both the 23rd and 24th centuries are capable of at least limited extrastellar communication.
But it is never clear if, say, an expedition left behind on a planet (either temporarily or long-term) might also have a little something extra left with them: maybe a satellite could be deposited in orbit. The satellite doesn't have to be a high-tech, high-powered Federation telecom buoy. Instead, it could be just an Expeditionary Package, to serve as a weather monitoring station, a local telecom relay so expedition team members don't loose touch over the horizon, and maybe a modest subspace booster to call nearby starships, space stations or expeditions if there are any in range. Such a "package" could be small, low-power and easy to store, set-up, handle and fix.
Specifically, consider that the Enterprise carried several hundred smallish "satellites" which were able to kill off the flying-rubber-vomit-monsters... remember? It's always seems ludicrous to me that the Enterprise would have the ability to manufacture that many, that quickly.
However... if they already had (as a standard part of the ship's mission, though never mentioned on-screen) a large inventory of satellites for the purpose you mention, well... it suddenly becomes practical that you could retrofit them with light sources, removing the transmitter parts and replacing them with easily-manufactured "sunlamp bulbs"... and voila.
A subspace relay satellite would inherently have a powerful power supply and some self-sufficiency built-in already.
The only thing that comes up, really, is the moral issue... how would it go over if you realized that the Federation is dropping what are, potentially, "spy satellites" into the neighborhood of every planet it visits?