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I have recently read this passage in The Fellowship of the Ring
He was kneeling in clear sunlight before the high seat.
A black shadow seemed to pass like an arm above him; it missed Amon Hen and groped out west, and faded.
Then all the sky was clean and blue and birds sang in every tree.
Could that shadow just pinpoint Frodo and take the One physically or just to point to the Nazgûl where Frodo was, at the moment he was wearing the ring?
Nowhere in Tolkien's Legendarium is it ever stated that the Ainur could simply reach out and take things from a distance. Further, Sauron used the Ithil-Stone to gaze out in search of the ring, which required great will power and could not been done constantly, it would not allow him to take physical possession of the ring.
I think the quote to answer your question is the one you provided (emphasis mine).
He was kneeling in clear sunlight before the high seat. A black shadow seemed to pass like an arm above him; it missed Amon Hen and groped out west, and faded. Then all the sky was clean and blue and birds sang in every tree.
The Fellowship of the Ring, Chapter 10 (The Breaking of the Fellowship)
Frodo's location, upon the seat of seeing on Amon Hen not only allowed him to gaze far, and in his gaze spot Barad-Dûr
The world seemed to have shrunk and fallen silent. He was sitting upon the Seat of Seeing, on Amon Hen, the Hill of the Eye of the Men of Númenor...
The Fellowship of the Ring, Chapter 10 (The Breaking of the Fellowship)
The text preceding your quote further suggests that Sauron was searching for the Ring, and wasn't able to physically grab it
Then at last his gaze was held: wall upon wall, battlement upon battlement, black, immeasurably strong, mountain of iron, gate of steel, tower of Adamant, he saw it: Barad-Dûr, Fortress of Sauron. All hope left him.
And suddenly he felt the Eye. There was an eye in the Dark Tower that did not sleep. He knew that it had become aware of his gaze. A fierce eager will was there. It leaped towards him; almost like a finger he felt it, searching for him. Very soon it would nail him down, know just exactly where he was. Amon Lhaw it touched. It glanced upon Tol Brandir – he threw himself from the seat, crouching, covering his head with his grey hood.
The Fellowship of the Ring, Chapter 10 (The Breaking of the Fellowship)
This passage suggests that although Sauron was searching for him,
"He knew it had become aware... It leaped towards him... he felt it"
This snippet discusses Sauron's search for the ring, with Sauron become aware of Frodo's gaze (implying he sees someone searching) and that Sauron's search "leaped towards him", and Frodo could feel his stare.
This leads me to conclude that he would've been unable to grab the ring directly from Frodo from afar, and was only able to pinpoint Frodo's location. To, in turn, send the Nazgûl in pursuit of the ring. In addition, the "Shadow" was, IMO, a literary representation for Sauron's gaze, and possibly a portrayal of his evil (as Blackwood's answer suggests)
I haven't found a discussion of this point by Tolkien, but remember he often uses "shadow" as a metaphor for something that is ominous or evil. A few examples are in the Ring Poem "In the Land of Mordor where the shadows lie." and in the title of the second chapter of The Lord of the Rings "The Shadow of the Past".
Sometimes the shadow is not just metaphorical, but can be seen as a cloud or mist. The passage quoted in the question mentions a "black shadow" that is "like and arm". We don't know if anyone other than Frodo sees this arm.
When the Ring is destroyed, we read:
‘The realm of Sauron is ended!’ said Gandalf. ‘The Ringbearer has fulfilled his Quest.’ And as the Captains gazed south to the Land of Mordor, it seemed to them that, black against the pall of cloud, 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.
The Lord of the Rings Book Six, Chapter 4: The Field of Cormallen
When Saruman is killed, we read:
To the dismay of those that stood by, about the body of Saruman a grey mist gathered, and rising slowly to a great height like smoke from a fire, as a pale shrouded figure it loomed over the Hill. For a moment it wavered, looking to the West; but out of the West came a cold wind, and it bent away, and with a sigh dissolved into nothing.
The Lord of the Rings Book Six, Chapter 8: The Scouring of the Shire
The last representations we see of both Sauron and Saruman show some resemblance to what Frodo sees in the quote in the question. There is no reason to suppose that they are actual pieces of the bodies of these Maia. Instead they seem to be symbolic of the departure of their "spirits". I believe that the quote in the question also shows a representation of Sauron's search for the ring.
Hypothetically, yes, if he was at full power - in the passage you describe, he was already defeated - and if he knew exactly where it was.
He was able to carry the Ring out of the wreck of Numenor.
Though reduced to ‘a spirit of hatred borne on a dark wind’, I do not think one need boggle at this spirit carrying off the One Ring, upon which his power of dominating minds now largely depended.
(Letters, 211)
He would have to travel there as a spirit, of course.
He might not have been willing to do so because a confrontation, even in spiritual form, would probably still be personally dangerous to him against one of the Noldor or Istari, or anyone, I suppose, who had been 'spiritually enlarged.' And as Denethor pointed out, Sauron was not a fan of personal confrontation (anymore?)
Denethor laughed bitterly. 'Nay, not yet, Master Peregrin! He will not come save to triumph over me when all is won. He uses others as his weapons. So do all great lords, if they are wise, Master Halfling. Or why should I sit here in my tower and think, and watch, and wait, spending even my sons? For I can still wield a brand.'
(LotR)
