The
Valar last intervened directly in
Middle-earth in full power (not even that) at the end of the
First Age to overthrow
Morgoth.
When
Númenor attacked
Valinor, they handed back their authority to
Eru, and the
Undying Lands were removed (however one is to imagine this) from being physically in Middle-earth; probably also to eliminate any possible repeat by any non-Elves in the future, remove the temptation, so to speak.
So I would think no direct intervention in Middle-earth by the Valar would occur again. But
Gandalf's being sent back as
The White has been widely interpreted as a direct intercession by Eru, if not directly in Middle-earth, except to send him back. So I would think that only some other direct intercession by Eru (though I simply can't guess what that would be) would follow on
Sauron's victory.
But this is very much fanfic (or alternative-world role-play) territory. JRRT made it a very close call, against seemingly impossible odds. But I'm pretty certain Sauron winning would never have taken place in
his Middle-earth.
One could, if one liked, deduce this from other writings, though most of them are only to be found in the twelve-volume "History of Middle-earth". The bit of the
Music of the Ainur being fate to practically all but humans, if I got that right (I'm not betting any sums ...) So Sauron's downfall could very well already have been foreseen in that music. That ominous "Second Prophecy of
Mandos" even deals with
events at the end of the world, unknown
ages in the future of the Third /
Fourth Age. So while the Valar may have been in the dark about much if not all of the future of
Men (and to a degree
Elves), Námo Mandos and
Manwë could have been in the picture, I would think, about the fate of their own kind, be it Sauron,
Saruman, Gandalf,
Radagast and those two
blue wizards (and the
Balrog of
Moria?).