According to the SW novel "Tarkin", he had very (very) strong suspicions but he kept them largely to himself:
Later, observing the rapport the Dark Lord shared with the
stormtroopers who supported him, and the technique he displayed in
wielding his crimson lightsaber, Tarkin grew more and more convinced
that his suspicions were right. Vader might very well be Jedi Knight
Anakin Skywalker, whom Tarkin had fought beside during the Clone Wars,
and for whom he had developed a grudging appreciation.
[later]
Sidious assumed that Tarkin had puzzled out that Vader had once been
Anakin Skywalker, under whom Tarkin had served during the war. Tarkin
may also have determined that Vader was a Sith. If so, it followed
that he accepted that Sidious was Vader’s dark side Master. But
Tarkin’s intuitions were important only in the sense that he never
revealed them and never allowed them to interfere with his own
ambitions.
He made a few half-hearted attempts to confirm whether it was Anakin behind the mask, but didn't push the issue:
“If we aren’t willing to do whatever is required,” he said finally,
“then we risk losing what we have been mandated to protect.”
The remark paraphrased something Skywalker had said to him following
the Citadel rescue. But it got no reaction from Vader beyond his
saying, “You misunderstand, Governor. As I said, we need to gather all
of them in our net.”