The rules for this are in the PHB, which mostly deals with PCs, which in turn usually can’t get multiple reactions (the only ways which can grant you multiple reactions tend to be limited to attacks of opportunity, and even those were only written after the PHB was released).
As such there is no language explicitly limiting the trigger to only be able to occur once in a round, even though I suspect that it is the intended reading. I find saying “the rules don’t explicitly say you can’t, so you can” not to be a useful guiding principle so I’d say the RAW isn’t clear on this. I personally would rule that no, you can only use your reaction to react to a trigger you set with Ready once because action economy wise that just makes a lot more sense, but I do not believe that a strict RAW reading of the section yields this as the answer.
RAW probably doesn’t matter in this case
If you decide your Marilith is picking up a longbow and shooting people with it you’re already moving beyond a literalist RAW reading, as of course shooting things with a longbow is not listed in the creature's statblock. This is not a big variation and perfectly in your purview of things to do as a DM, but you have to decide on what the consequences here are. For example, does it still get a Multiattack? If so, how many? Maybe it only gets 3 shots because it needs two arms for each bow? Similarly, the multi reaction thing was very probably themed after it using its many arms with one handed weapons; Does that still even apply if it uses bows instead? Should it still work even if it is using only one bow?
At this point what you need to realise that you are effectively homebrewing a slightly different creature with a slightly different statblock, and it’s on you as a DM to figure out how it should work. There isn’t that much helpful RAW guidance for you here. NPCs aren’t bound by quite the same rules as PCs to begin with, you should just make sure that whatever you do it makes sense in fiction so your players can know what to expect from your description of events.