They weren't either. The colonists of Williamsburg weren't Puritans. Puritans were both a religious and a political movement back in England. There were three political factions that arose out of the reign of Henry VIII which still existed at the time that both Jamestown and Plymouth were founded. The conservatives were called the "Papists", the pro-Catholic faction that thought that Henry VIII's split from the Catholic Church was wrong and wanted to bring the Church of England back under the authority of the Pope. Then there were the more progressive "Protestants", supporters of the King's position who protested the authority of the Catholic Church, and thought that Henry VIII was right. And then there were the more left-wing (politically, though not always in their religious interpretations) "Puritans" who thought that the King hadn't gone far enough. He had simply replaced one wrong-headed political body under the authority of the Pope with another wrong-headed political body under the authority of the King, and they wanted to purify the church from the interference of governmental politics altogether. The Puritans called themselves the "Puritans" while the King actually called them the "Dissenters" or the "Separatists".
The founders of Virginia weren't part of the Puritan movement. They were part of the Protestant movement. The church parish established in Jamestown in 1607, and then later established in the Middle Plantation (Williamsburg) in 1632 was an Anglican church--that is, a parish of the Church of England, with the King as the head of the church. This is opposed to the Congregationalist churches established up in Plymouth by members of the Puritan political/religious faction, which did not have the King as the head of their church.
And since they weren't Puritans, the colonists of Jamestown and Williamsburg can't really be called "Pilgrims", either, because a "pilgrim" is a person who travels to a holy place--at least, that's what its meaning was at the time of the founding of Virginia and Plymouth. It wasn't until much later that "Pilgrim" became synonymous with the founders of Plymouth. The people of Virginia never used the term "Pilgrim" to describe themselves, nor did they come to Virginia for religious reasons.
For that matter, the Puritans in Plymouth didn't ever really call themselves "Pilgrims", either. Governor William Bradford wrote On Plymouth Plantation in c.1630 where he quoted the Bible verse Hebrews 11:13, identifying the colonists with the "pilgrimes" mentioned in that Bible verse, but that was about the extent of it.
It wasn't until 1800 when Bradford's term "Pilgrim" was resurrected that it came into widespread use to describe the founders of Plymouth, as detailed in the study "The Term Pilgrim Fathers" by Albert Matthews. It was more of a 19th Century phenomenon, when a celebration was held in Boston as "The Feast of the Sons of the Pilgrims". In the years immediately prior, in 1797-98, the same observance had been billed as the "Celebration of our Forefathers at Plymouth".
SOURCES and FURTHER READING:
You can read more on the religious history of the colonists of Williamsburg in the article "Historic Parishes of America: Bruton Parish" by George Carrington Mason.
Here are some 17th Century sources talking about Protestants, Puritans, and Papists: 1, 2, 3, 4.
And here's a 1732 book called History of the Puritans which talks extensively about the political factions of Papists, Puritans, and Protestants, and how Queen Elizabeth I found a middle way reconciling the Papists and Puritans on either side of the political wing she represented, the Protestants.
And here are a few sources of the era displaying how the pro-royal Protestants often called the "Puritans" by the name of the "Dissenters", "Separatists", or "Non-Comformists": 1, 2, 3.
More recently, the book Papists, Protestants and Puritans 1559-1714 by Diane Newton, published in 1998, details the history of these political/religious factions as well.