The "wig party" people search for is usually the Whig Party. It was a political party, not a hairstyle party.

In the United States

The Whig Party formed in the 1830s in opposition to Andrew Jackson and the Democratic Party. It included figures such as Henry Clay and was active before the rise of the Republican Party.

Why the name sounds like wig

Whig and wig sound the same, but Whig comes from political history, including British anti-royal associations.

Bottom line

The Whig Party was a nineteenth-century political party. If you typed "wig party," you probably meant Whig.