In Harry Potter and the Deathly Hallows, during the part where Nagini appears in Malfoy Manor with the Death Eaters, there is a line which says:
To Malfoy's left, his wife made an odd stiff nod, her eyes averted from Voldemort and the snake.
Which makes me wonder whether it is possible for Voldemort to perform Legilimency via the snake/Horcrux?
While he was still noncorporeal, he possessed snakes, but animals weren't suited to perform magic, even while holding a piece of the Dark Lord's soul.
“I sometimes inhabited animals – snakes, of course, being my preference – but I was little better off inside them than as pure spirit, for their bodies were ill-adapted to perform magic … and my possession of them shortened their lives; none of them lasted long …” - Harry Potter and the Goblet of Fire, Chapter 33 (The Death Eaters)
Despite knowing nonverbal and wandless magic, he still described it as being almost no better possessing animals as being a spirit, because they weren't suited to perform magic. Therefore, while not conclusively disproven, it seems unlikely that he could use Nagini to perform Legilimency. Although he never addressed whether he could perform Legilimency while possessing animals, he does say they're not well-suited to magic, and the only difference between them and Nagini is that she would have two pieces of his soul if he possessed her, and she was stronger than an ordinary snake. However, that doesn't necessarily mean that she would be suitable to use to perform magic, including Legilimency.
Nagini isn't particularly capable of doing magic either.
When the Dark Lord possessed her to attack Arthur Weasley, she didn't perform any magic. She bit him, but didn't use any magic - even though the Dark Lord was possessing her at the time.
“But the man was stirring … a silver Cloak fell from his legs as he jumped to his feet; and Harry saw his vibrant, blurred outline towering above him, saw a wand withdrawn from a belt … he had no choice … he reared high from the floor and struck once, twice, three times, plunging his fangs deeply into the man’s flesh, feeling his ribs splinter beneath his jaws, feeling the warm gush of blood … The man was yelling in pain … then he fell silent … he slumped backwards against the wall … blood was splattering on to the floor …” - Harry Potter and the Order of the Phoenix, Chapter 21 (The Eye of the Snake)
In addition, she didn't perform any magic while she was inside Bathilda Bagshot, in Bathilda's house with Harry and Hermione. Her not using magic was obvious and out of place enough that Harry noticed it and presumed it was because Bathilda was a bit senile.
“She seemed to have forgotten that she could do magic too, for she lit the candles clumsily by hand, her trailing lace cuff in constant danger of catching fire.
‘Let me do that,’ offered Harry, and he took the matches from her. She stood watching him as he finished lighting the candle stubs that stood on saucers around the room, perched precariously on stacks of books and on side tables crammed with cracked and mouldy cups.” - Harry Potter and the Deathly Hallows, Chapter 17 (Bathilda's Secret)
There's never been an instance in the books that I know of where Voldemort has used legilimency in such a manner. However, that's not to say it isn't possible.
Nagini is a horcrux. This we know. Because of this, she has a connection with Lord Voldemort that grants her certain benefits, and also allows Voldemort to possess her on occasion, as he did in order to attack Arthur Weasley (coincidentally, this is why Harry also saw the attack).
Legilimency is an art that allows a wizard to enter the mind of another. Voldemort is well known for his skill in legilimency and is certainly one of, if not the, best legilimens who ever lived. A number of Wizards can perform legilimency nonverbally and wandlessly, simply by looking into the eyes of another person. Notably, this has been performed by Snape, Dumbledore and of course Voldemort.
So, if Voldemort were possessing Nagini through their connection, it is possible that he could nonverbally and wandlessly perform legilimency by looking into another's eyes through Nagini's eyes.
Though as I said, this hasn't been shown in the books or movies so it's mainly speculation based on what we have seen.
