In Middle-earth, does there exist magic that causes the user to gain influence over another's body and actions?
At the same time, are there catalysts in Middle-earth where such a spell could be bound to?
Yes there is such magic. Sauron uses some on Pippin during their conversation in Orthanc.
From The Two Towers, Book III, Chapter 11, The Palantir:
Tell Saruman that this dainty is not for him. I will send for it at once. Do you understand? Say just that!
And the next thing Pippin says is
It is not for you, Saruman! I will send for it at once. Do you understand? Say just that!
In addition to that encounter, Sam uses the One Ring to control some orcs (From The Return of the King, Book VI, Chapter 1, The Tower of Cirith Ungol)
For what it saw was not a small frightened hobbit trying to hold a steady sword: it saw a great silent shape, cloaked in a grey shadow, looming against the wavering light behind; in one hand it held a sword, the very light of which was a bitter pain, the other was clutched at its breast, but held concealed some nameless menace of power and doom.
For a moment the Orc crouched, and then with a hideous yelp of fear it turned and fled back as it had come.
Frodo used it to control Gollum (from The Return of the King, Book VI, Chapter 3, Mount Doom):
Then suddenly, as before under the eaves of the Emyn Muil, Sam saw these two rivals with other vision. A crouching shape, scarcely more than the shadow of a living thing, a creature now wholly ruined and defeated, yet filled with a hideous lust and rage; and before it stood stern, untouchable now by pity, a figure robed in white, but at its breast it held a wheel of fire. Out of the fire there spoke a commanding voice.
Begone, and trouble me no more! If you touch me ever again, you shall be cast yourself into the Fire of Doom.’
The crouching shape backed away, terror in its blinking eyes, and yet at the same time insatiable desire.
Then the vision passed and Sam saw Frodo standing, hand on breast, his breath coming in great gasps, and Gollum at his feet, resting on his knees with his wide-splayed hands upon the ground.