Did JKR explain how the Schools founding Wizards got the castle? Did they charm muggles into building it, did they buy it from the Romans, or is there another answer?
The wizarding community doesn't seem to have any problem building structures on demand (such as for the Quiditch World Cup). The most likely explanations are:
Animated workers similar to the animated suits of armor from the end of The Deathly Hallows
I would also guess that there are skilled craftsman (equivalent of muggle general contractors) who have developed a repertoire of spells, tools, employees and/or house elves. These folks would be well known and the "go to" guys for major projects.
It was built by the magical community.
There are hints in the canon that magical folk had a hand in its design, which strongly implies that it was a new construction, not an existing building that was acquired:
An article about The Wizarding World of Harry Potter amusement park mentions an architect:
The front hall also boasts statues of the One-eyed Witch and the Architect of Hogwarts (unnamed, he is also seen frequently in the films; he's the gold/bronze statue holding a model of Hogwarts in his hand).
Since the architect is unnamed, it seems reasonable to assume this is probably not one of the four named founders of the school.
Quoting Rowena Ravenclaw’s Famous Wizard Card:
Co-founder of Hogwarts. Gave her name to one of the four Hogwarts houses. It's thought that she came up with the ever-changing floor plan. She's best remembered for her intelligence and creativity.
An ever-changing floor plan would have to be constructed with at least some magical assistance. There’s just no way Muggles alone could have built in, so a witch or wizard must have been involved.
More generally, I agree with @Donald.McLean that building Hogwarts seems well within the capabilities of magic that we’ve seen. Although we don’t have any specifics about the people who put the bricks together, building Hogwarts is consistent with the known abilities of magical folk.
The founders built the castle themselves.
When Professor Binns tells his History of Magic class how Hogwarts was founded, he says that the four founders of Hogwarts built the castle together far from Muggle eyes.
“You all know, of course, that Hogwarts was founded over a thousand years ago – the precise date is uncertain – by the four greatest witches and wizards of the age. The four school houses are named after them: Godric Gryffindor, Helga Hufflepuff, Rowena Ravenclaw and Salazar Slytherin. They built this castle together, far from prying Muggle eyes, for it was an age when magic was feared by common people, and witches and wizards suffered much persecution.”
- Harry Potter and the Chamber of Secrets, Chapter 9 (The Writing on the Wall)
They did not buy it, steal it, or make Muggles build it - they built it themselves.
Hogwarts was likely built using magic.
The founders would have been helped in their building of Hogwarts by their magical abilities. Isolt Sayre, one of the founders of Ilvermorny, built an entire stone cottage in an afternoon. Presumably the founders of Hogwarts built the Hogwarts castle in this way, its construction sped along greatly by their magic.
James even helped Isolt construct a stone house on the top of Greylock, providing a workable design, having been a stonemason in England, which Isolt made a reality in the space of an afternoon. Isolt christened her new home ‘Ilvermorny’ after the cottage in which she had been born, and which Gormlaith had destroyed.
- Ilvermorny (Wizarding World website)
Construction using magic is seen to be more efficient. In the Fantastic Beasts and Where to Find Them movie, Aurors quickly repair the buildings of New York with magic.
“Aurors move through the streets, performing Repairing Charms to rebuild the city: buildings and cars are reconstructed and streets are returned to normal.”
Fantastic Beasts and Where to Find Them: The Original Screenplay
It's likely that wizards use similar methods to those seen in Fantastic Beasts to build new buildings.