This is an interesting question but there is no explicit answer in either the Silmarillion or LotR. However, given an understanding of how Tolkien saw the Ainur (such as Sauron), the answer is pretty clear.
To start, I'd be very careful about trying to make a sharp distinction between physical and metaphysical in Tolkien's world, since it does not seem to exist there. The 'higher' a being is, the more their 'physical' appearance is simply a function of their nature and derived from it. The Ainur didn't have or need sense organs to perceive the world, and when they assumed forms possessing sense organs it was for the purpose of appearing in the world, not because they needed them. The physical forms of the Ainur reflected their personalities and even their whims, but not their needs.
From the Silmarillion:
Now the Valar took to themselves shape and hue [...] Moreover their shape comes of their knowledge of the visible World, rather than of the World itself; and they need it not, save only as we use raiment, and yet we may be naked and suffer no loss of our being. [...] Therefore the Valar may walk, if they will, unclad, and then even the Eldar cannot clearly perceive them, though they be present. But when they desire to clothe themselves the Valar take upon them forms some as of male and some as of female; for that difference of temper they had even from their beginning, and it is but bodied forth in the choice of each, not made by the choice, even as with us male and female may be shown by the raiment but is not made thereby. But the shapes wherein the Great Ones array themselves are not at all times like to the shapes of the kings and queens of the Children of Ilúvatar; for at times they may clothe themselves in their own thought, made visible in forms of majesty and dread.
and later
here is told in brief their likenesses, such as the Eldar beheld them in Aman. But fair and noble as were the forms in which they were manifest to the Children of Ilúvatar, they were but a veil upon their beauty and their power. And if little is here said of all that the Eldar once knew, that is as nothing compared with their true being,
Of Melkor (Morgoth):
For he was yet as one of the Valar, and could change his form, or walk unclad, as could his brethren; though that power he was soon to lose for ever.
So the Ainur (of which Sauron also was one) were, in the beginning, creatures for whom a physical form was like clothing for people in a tropical climate: Mostly a social convenience.
Melkor and Sauron lost the ability to take on any shape, not because they lost "powers" extrinsic to themselves, but because in their evil desires and because of the destructive deeds they did, they diminished themselves. They diminished their own essential being, and the limitations in their physical forms reflected that.
When Morgoth destroyed the Two Trees and escaped to Middle-earth he took on a dreadful aspect to physically dominate Middle-earth. And that choice costs him the ability to walk "unclad". After causing the Fall of Númenor, Sauron became unable to take on a "fair form". And when he lost the Ring to Isildur (into which he had poured part of himself, part of his vitality), he became unable to take on any physical form at all for thousands of years. What happens when the Ring is destroyed:
[Gandalf says that with the destruction of the Ring, Sauron will lose] the best part of the strength that was native to him in his beginning, and all that was made or begun with that power will crumble, and he will be maimed for ever, becoming a mere spirit of malice that gnaws itself in the shadows, but cannot again grow or take shape.
Once again, his diminished shape will reflect his diminished spirit.
[And when the Ring goes into the fire] there rose a huge shape of shadow, impenetrable, lightning-crowned, filling all the sky. Enormous it reared above the world, and stretched out towards them a vast threatening hand, terrible but impotent: for even as it leaned over them, a great wind took it, and it was all blown away, and passed; and then a hush fell.
Sauron's embodiment (like Morgoth's, and like all the Ainur's) is a reflection of his spirit, his vitality, not the source of it. When his spirit is gravely weakened by the destruction of the Ring his body blows away into nothingness.
Bottom line: Sauron, being an Ainu, did not rely on the senses of his body. His body was to him as clothing is to us: Perhaps useful, but not essential.
Consequently, the Eye of Sauron is something that Sauron possesses that is independent of his body. What it is exactly, we don't know of course. (Tolkien was far too great a writer to try to spell out the details of the supernatural.) It seems likely that Sauron was simply using the perception that all the Ainur had. For example, Manwë and Varda see (and hear) far beyond actual eyesight (or earshot):
When Manwë [...] ascends his throne and looks forth, if Varda is beside him, he sees further than all other eyes, through mist, and through darkness, and over the leagues of the sea. And if Manwë is with her, Varda hears more clearly than all other ears the sound of voices that cry from east to west [...].
and even Frodo and Aragorn seem to be temporarily gifted with this perception at the Seat of Seeing on Amon Hen
At first he could see little. He seemed to be in a world of mist in which there were only shadows: the Ring was upon him. Then here and there the mist gave way and he saw many visions: small and clear as if they were under his eyes upon a table, and yet remote. There was no sound, only bright living images. The world seemed to have shrunk and fallen silent. Eastward he looked into wide uncharted lands, nameless plains, and forests unexplored. Northward he looked, and the Great River lay like a ribbon beneath him, and the Misty Mountains stood small and hard as broken teeth. Westward he looked and saw the broad pastures of Rohan; and Orthanc, the pinnacle of Isengard, like a black spike. Southward he looked, and below his very feet the Great River curled like a toppling wave and plunged over the falls of Rauros into a foaming pit; a glimmering rainbow played upon the fume. And Ethir Anduin he saw, the mighty delta of the River, and myriads of sea-birds whirling like a white dust in the sun, and beneath them a green and silver sea, rippling in endless lines.
Whatever the source of Sauron's ability to perceive afar, it wasn't based on whatever physical eye he possessed. Neither he nor anyone else who had that ability needed flaming eyeballs to use it.
Note also that people repeatedly feel the presence of powerful evil creatures near them. (I'll spare the quotes, because there are many.) They feel horror and fear and this is entirely due to the nearby presence of those powers. There is no reason to think that this fear is mediated by anything physical, and it is certainly not mediated by the sight of the evil creature, as they are frequently explicitly stated to be unseen. How much more Sauron?
It seems most likely that the people who were under the gaze of the Eye of Sauron perceived the combination of present evil, the force of Sauron's terrible will and the pressure of whatever power of perception Sauron possessed. They perceived this as a fiery eye. Sauron's Eye was more, then, than a metaphor, but not an actual eyeball in Sauron's head.
(Of course it's possible that the final form that Sauron possessed may not have been simply oversized-though-otherwise-normal. He may have had flaming eyes for all we know — his form is not described. But we can be confident that if he had flaming eyes or a flaming eye in his head, it was there to scare the troops and not as the engine of his perception.)
In the end we don't need to assume his body had flaming eyes to account for the terrible impact he had on people.
In summary:
The Ainur existed independently of their bodies. And even when bound to a body, it was because they had diminished their beings. Their bodies 'reflected' their being and were not the source of it.
The Ainur (and occasionally others) could see beyond the natural visual range using some sort of mental or spiritual — not physical, anyway — perception.
The presence of evil creatures (even weak evil creatures) could be sensed by anyone so unlucky to be near to them or watching them.
The Eye of Sauron was how people perceived being perceived by Sauron and felt the edge of his terrible will. If it had a manifestation in his physical body, it was a reflection of his will and his perception, not the cause of it.
