I'm answering the question as per your approval of winwaed's comment:
Or do you mean when it was realized that they were physical objects with Keplerian orbits?
I've looked at a bunch of sources, and they seem to indicate that the honor of discovering the comets are celestial bodies goes to Tycho Brahe, Isaac Newton and Edmund Halley.
For centuries, scientists thought comets traveled in the Earth's atmosphere, but in 1577, observations made by Danish astronomer Tycho Brahe revealed they actually traveled far beyond the moon.
It was Isaac Newton who expanded upon this to study their orbits:
Isaac Newton later discovered that comets move in elliptical, oval-shaped orbits around the Sun, and correctly predicted that they could return again and again.
The Encyclopedia Britannica tells a slightly different story (though with the same conclusion). Apparently, Aristotle first theorized about comets, claiming that they were just strange atmospheric phenomena. Some disagree - notably, the philosopher Seneca, who could lay claim to being the first person to identify comets as celestial bodies - but, of course, Aristotle prevailed. But Brahe came to the rescue:
Finally, during the 16th century the Danish nobleman Tycho Brahe established critical proof that comets are heavenly bodies. He compared the lack of diurnal parallax of the comet of 1577 with the well-known parallax of the Moon (the diurnal parallax is the apparent change of position in the sky relative to the distant stars due to the rotation of Earth). Tycho deduced that the comet was at least four times farther away than the Moon, establishing for the first time that comets were heavenly bodies.
Go to the next page in the Encyclopedia Britannica article, though, and you'll find that Newton and Halley did important work on the orbits of comets.
This page completely leaves out Brahe (!) and credits Newton and Halley with dominating the field of studying comets:
The next real progress was to show that comets obeyed the same natural laws as other bodies in the heavens, and this had to wait until someone--Sir Isaac Newton--determined what those laws were. Sir Edmund Halley, a close friend of Newton, tamed the mystery of comets and broke the prediction barrier. The popular recent theory of Kepler was that comets travel in straight lines. This seemed to allow for the way comets' tails changed orientation to point anti-sunward as they moved. But Newton's finding that heavenly bodies moved on closed, gravitationally bound orbits appealed to Halley.
And, later on,
Halley did some very difficult calculations, and discovered that a comet with an elliptical orbit and period of 76 years could account in detail for the sightings of 1607, 1531, and 1456.
So perhaps you can credit this to Brahe, Newton, and Halley.
By the way, this page is exemplary, but it repeats a lot of the information stated above. That said, it's a fairly reliable source.