Wordorigins.org

Gyp or gip, pronounced with a soft g, got its start as a derogatory term for Gypsy.

The term dates to the mid-19th century. From Gipsey Davy, found in Francis Child’s English and Scottish Ballads, and written sometime before 1840:

There was a gip came o’er the land.

The sense of a thief or swindler is an American one. Gip is glossed as a thief in George Matsell’s Rogue’s Lexicon of 1859. The use of the term to mean a fraud dates to the early 20th century. From Jackson and Hellyer’s 1914 A Vocabulary of Criminal Slang:

Gyp...the act of short-changing; a defrauding by substitution; an action that belies a professed sincerity.

The verb is glossed in the 1880 Century Dictionary.

(Source: Historical Dictionary of American Slang)