Gov. Eric Greitens of Missouri was accused on Friday of illegally obtaining a donor list from a charity he founded and using it to raise money for his 2016 campaign, a new blow to Mr. Greitens in a deepening political and sex scandal that has threatened his grip on the governor’s office.

Kimberly Gardner, the St. Louis circuit attorney, charged Mr. Greitens with one felony count of tampering with computer data in connection with the donor list.

Mr. Greitens, a Republican who was first elected governor in the 2016 race, already was facing a different felony charge, invasion of privacy. In January, he admitted that he had a sexual relationship with his former hairdresser while he was married. The woman has told a legislative committee that Mr. Greitens took a nude or seminude picture of her without her consent, then threatened to make the image public if she told anyone about their affair.

The new felony charge dates back to actions prosecutors say took place in April 2015, when Mr. Greitens, a former Navy SEAL, was running for governor and serving on the board of directors for The Mission Continues, a veterans charity that he founded. According to the charging document, Mr. Greitens directed an employee of the charity, identified only as “K.T.,” to take a list of donors from the charity.