A Level 18 Way of Shadow monk wants to Ready his action to use his reaction to cast Darkness on the tail feather of an arrow shot by one of his allies in mid-flight. Is this sort of maneuver doable, or would it require some sort of check dictated by the DM in order to do so?
It depends on how the DM interprets the rules
If you’re unsure when a reaction occurs in relation to its trigger, here’s the rule: the reaction happens after its trigger completes, unless the description of the reaction explicitly says otherwise.
As Slagmoth points out, Sage Advice (page 10) also says:
Can a readied action occur during an enemy’s triggering action, such as between its first and second attacks when it uses Multiattack?
A readied action occurs immediately after its trigger. If you defined the trigger as an attack, your action happens after that attack.
I think the question is then: when does the attack end? It is important to note that the rules to not define a travel time for projectiles. One could argue that it's after the arrow has landed/hit. Or it could be after the arrow has left the bow, since the attacker then could move on to do something else (conceivably before the arrow has landed, given a long travel distance).
The rules do not (I think) provide a definitive answer to this question, so the DM will need to adjudicate and be aware that allowing reactions between projectile departure and landing like this might have unintended consequences in other situations (such as allowing catapult to bounce a projectile back at the attacker)
If you wanted to argue semantics, you could define the trigger not as the attack but as the arrow being positioned a few feet from the attacker. This leans on the description of using a readied action:
First, you decide what perceivable circumstance will trigger your Reaction. Then, you choose the action you will take in response to that trigger.
However, the XGtE and Sage Advice rule from above further refine and narrow what a "perceivable circumstance" can be in the case of a weapon attack. As such, this is more an attempt to use a loophole in the wording than a solid, rules-based argument.
A stronger argument (I think) would be to lean on Rule Zero From page 6 of the PHB:
•The DM describes the environment.
•The players describe what they want to do.
•The DM narrates the results of the adventurers' actions.
And from the DMG:
The D&D rules help you and the other players have a good time, but the rules aren't in charge. You're the DM, and you are in charge of the game. That said, your goal isn't to slaughter the adventurers but to create a campaign world that revolves around their actions and decisions, and to keep your players coming back for more!
Using these sections of the rule books, the DM could allow a novel use of Darkness like you describe.
This is entirely doable. The Darkness spell says:
If the point you choose is on an object you are holding or one that isn’t being worn or carried, the darkness emanates from the object and moves with it.
As the ready action can be used to cast spells, you may specify that you cast the spell as soon as the arrow is fired. Of course, you would have to specify which spell you would use on the turn you take the Ready action, and the circumstances in which you would cast it.
You should not have to make a check to do so.