the lord of the rings - If Sauron had won, would the Valar have intervened?

In the First Age, when the Elves first awoke and were terrorized by Morgoth, the Valar came to help as soon as they learned of it, defeated and captured Morgoth.

When the Noldor went against the will of the Valar and followed Morgoth to Middle-earth, and were eventually defeated by him, the Valar listened to Eärendil's pleas and began the War of Wrath, defeated and captured Morgoth.

So if Gandalf and Frodo had failed, would Sauron really have had complete dominance forever? Or would the Valar have intervened again? If not, why? Is it because in Arda Marred, the Valar are not able and/or willing to directly intervene anymore, and sending the Istari was all they could do?

Are there any statements in Tolkien's writings about this?

Answer

Most likely not.

The Valar didn't really directly intervene since bringing the Elves to Valinor and the first capture of Melkor; they were essentially powerless during the events leading to the capture of the Silmarils and the Flight of the Noldor, they sent a lesser representative (Eonwe) to lead the War of Wrath (the relevant chapter of the Silmarillion is quite careful to always mention "the host of the Valar" rather than "the Valar", and has Eonwe cast in a quite definite decision making/judgemental role), and even the downfall of Numenor was not an intervention by them - it was a case of "for that time the Valar laid down their government of Arda".

There was also the small matter of the Music of the Ainur - in particular the fact that the initial vision was taken away before the end: "and some have said that the vision ceased ere the fulfilment of the Dominion of Men and the fading of the Firstborn; wherefore, though the Music is over all, the Valar have not seen as with sight the Later Ages or the ending of the World". In other words, it's quite clearly inferred that by the time we get to the end of the Third Age and the War of the Ring, the Valar very probably just do not know what Eru's intention actually is - intervention on their part would have been a huge risk; they may be going against the intended destiny of the world, and for all they knew a victory for Sauron may have even been the event that precipitated the Dagor Dagorath. Big mistake to try prevent that!

(All references the published Silmarillion.)