That is a provocative question Laurelin.

It is my feeling that the Valar were very pleased with their capture of Melkor in the old days and went into some form of hiatus or another. In the aether of my imagination I perceive that the Valar were hard put to The Keeping of Melkor in the Void (being the most powerful of the Valar), or that they were saddened by the breaking of the world and the destruction of Beleriand and all those other original territories in the War of Wrath, forsaking the remaining peoples of Middle-earth to care for the world themselves.

I feel that this kind of speculation is thoroughly entertaining and for the imaginative, a source of nourishment but where I see more depth in promise of furthering the realms of Middle-earth most only see a dead end.

I say "Dare to Dream,"
If there was one embodiment of the Valar that still came into play in LOTR or War of the Ring it was the Eagles! They were said to be the heralds of Manwe. Perhaps this describes a bit of the relationship between Gandalf and Gwaihir or the Eagles in general....

Besides; if it was written of at all someone will come forth with it soon....party on!

This little thread intrigued me when my friend Laurelin brought it up.
It took me a few days to get in touch with a connection and sort it through.

I was intrigued because I know that works like "The Lord of The Rings" (which is a part of the War of The Rings) had deep roots in works like "The Silmarillion." So I looked for thematic threads in which the Ainur were part of the mythos of Middle-earth. And it took me to recognizing that the way these creatures were made manifest was that they were the creations of Eru. And the first thing Eru had them do was make music to a theme he presented.
So I think another piece could be added if we look to the Ainulindale ~ "The Music of the Ainur".
"And he spoke to them, propounding to them themes of music; and they sang before him, and he was glad. But for a long while they sang only each alone, or but few together, while the rest hearkened; for each comprehended only that part of the mind of Ilúvatar from which he came, and in the understanding of their brethren they grew but slowly. Yet ever as they listened they came to deeper understanding, and increased in unison and harmony."
(The Silmarillion, Ainulindale ~ "The Music of the Ainur", pg 15).

The fact they ever grew deeper in understanding seems to say that eventually they would recognize and reject the disharmony of Melkor and those Maiar that he had seduced.
Exactly what their response would be was not disclosed to them yet.

"Then Ilúvatar said to them: 'Of the theme that I have declared to you, I will now that ye make in harmony together a Great Music. And since I have kindled you with the Flame Imperishable, ye shall show forth your powers in adorning this theme, each with his own thoughts and devices, if he will. But I will sit and hearken, and be glad that through you great beauty has been wakened into song.' (ibid, pg 15)

But the question was; "Has anybody ever wondered what the inhabitants of Valinor were doing/thinking while the War of the Ring was going on?"
And I think to that question we also can refer to the Ainulindale to see a pattern of their (the Ainur) response.
"Then Ilúvatar arose, and the Ainur perceived that he smiled; and he lifted up his left hand, and a new theme began amid the storm, like and yet unlike to the former theme, and it gathered power and had new beauty. But the discord of Melkor rose in uproar and contended with it, and again there was a war of sound more violent than before, until many of the Ainur were dismayed and sang no longer, and Melkor had the mastery."(ibid, pg 16)
The Ainur no longer sang ... so their actions ceased while Melkor and his minions dominated, at least in the first singing.
If we hypothesize that The Music of the Ainur is what makes up Arda and its unfolding history then there may be spaces where Iluvatar alone confronts the disharmony of Melkor and his minions.

Now here is where my own imagination really is my own projection ... that is that part of Iluvatar's third intervention in the Music of the Ainur ..... is  "The War of The Ring."

"Then again Ilúvatar arose, and the Ainur perceived that his countenance was stern; and he lifted up his right hand, and behold! a third theme grew amid the confusion, and it was unlike the others. For it seemed at first soft and sweet, a mere rippling of gentle sounds in delicate melodies; but it could not be quenched, and it took to itself power and profundity. And it seemed at last that there were two musics progressing at one time before the seat of Ilúvatar, and they were utterly at variance. The one was deep and wide and beautiful, but slow and blended with an immeasurable sorrow, from which its beauty chiefly came. The other had now achieved a unity of its own; but it was loud, and vain, and endlessly repeated; and it had little harmony, but rather a clamorous unison as of many trumpets braying upon a few notes. And it essayed to drown the other music by the violence of its voice, but it seemed that its most triumphant notes were taken by the other and woven into its own solemn pattern." (ibid, pgs 16-17)

So part of what I perceive is that while we are in the "music stage" of the construct of Arda  the Ainur do nothing and fall silent.
But Iluvatar says;
"Mighty are the Ainur, and mightiest among them is Melkor; but that he may know, and all the Ainur, that I am Ilúvatar, those things that ye have sung, I will show them forth, that ye may see what ye have done. And thou, Melkor, shalt see that no theme may be played that hath not its uttermost source in me, nor can any alter the music in my despite. For he that attempteth this shall prove but mine instrument in the devising of things more wonderful, which he himself hath not imagined." (ibid, pg 17)
In other words ... don't sweat it ... the devices and crap that Melkor stirs up is powerless ... I and my Flame Imperishable ... we are the real thing ...
What the Bear thinks is, that in the music, "The War of The Rings" is played out. And Iluvatar calls the tune.

The Ainulindale goes on to say that, at first, the Ainur didn't get it.
"Then the Ainur were afraid, and they did not yet comprehend the words that were said to them; and Melkor was filled with shame, of which came secret anger. But Ilúvatar arose in splendor, and he went forth from the fair regions that he had made for the Ainur; and the Ainur followed him." (ibid, pg 17)

In their fear the Ainur don't contribute to the various themes ... because they do not understand ... But Iluvatar takes them to a place where they can see what their music makes and how they are part of Iluvatar's mighty creation ...
    "But when they were come into the Void, Ilúvatar said to them: 'Behold your Music!' And he showed to them a vision, giving to them sight where before was only hearing; arid they saw a new World made visible before them, and it was globed amid the Void, and it was sustained therein, but was not of it. And as they looked and wondered this World began to unfold its history, and it seemed to them that it lived and grew. And when the Ainur had gazed for a while and were silent, Ilúvatar said again: 'Behold your Music! This is your minstrelsy; and each of you shall find contained herein, amid the design that I set before you, all those things which it may seem that he himself devised or added. And thou, Melkor, wilt discover all the secret thoughts of thy mind, and wilt perceive that they are but a part of the whole and tributary to its glory.'
    And many other things Ilúvatar spoke to the Ainur at that time, and because of their memory of his words, and the knowledge that each has of the music that he himself made, the Ainur know much of what was, and is, and is to come, and few things are unseen by them. Yet some things there are that they cannot see, neither alone nor taking counsel together; for to none but himself has Ilúvatar revealed all that he has in store, and in every age there come forth things that are new and have no foretelling, for they do not proceed from the past. And so it was that as this vision of the World was played before them, the Ainur saw that it contained things which they had not thought. And they saw with amazement the coming of the Children of Ilúvatar, and the habitation that was prepared for them; and they perceived that they themselves in the labor of their music had been busy with the preparation of this dwelling, and yet knew not that it had any purpose beyond its own beauty. For the Children of Ilúvatar were conceived by him alone; and they came with the third theme, and were not in the theme which Ilúvatar propounded at the beginning, and none of the Ainur had part in their making. Therefore when they beheld them, the more did they love them, being things other than themselves, strange and free, wherein they saw the mind of Ilúvatar reflected anew, and learned yet a little more of his wisdom, which otherwise had been hidden even from the Ainur." (ibid, 17-18)

So in the "song" or "time and space" of the "War of The Rings" certain of the Valar did join in ... on either side ...
It is up to my own individual conjecture but here are a few examples of how some of the Valar contributed :

• Yavanna ~ the Ents, Tom Bombadil and Goldberry.
• Ulmo ~ the arrival of  Aragorn up the river at the battle of Pelennor Fields.
• Mandos ~ the Oath Breakers that came from the Paths of the Dead
• Manwe ~ The giant eagles
• Orome ~ Shadowfax
• Varda ~ the phial from Galadriel
• Etc and etc ...

So I think I will leave it with you to draw the many other examples of the Valar participation in this struggle "War of The Rings."
 
So what do you think?