Many officeholders in the United States are required to take an oath of office. How would a mute person do this?
By whatever means that person communicates. ASL (sign language), sign a piece of paper with the words, write the words and sign them...
The original law from the U.S. Constitution is for the president, Article II, Section 1:
Before he enter on the Execution of his Office, he shall take the following Oath or Affirmation:-"I do solemnly swear (or affirm) that I will faithfully execute the Office of President of the United States, and will to the best of my Ability, preserve, protect and defend the Constitution of the United States."
It does not say anything about speaking those words aloud. So in whichever way "he" is able to take the oath should suffice.
Ask Amanda Folendorf
She appears to be the only deaf elected "chief executive" (mayor, governor, president, etc.) in the US. I can't find any resources which say whether she swore the oath of office verbally, via interpreters, or another way. However, she does have a team of interpreters to navigate her daily duties, so I would presume they performed during her swearing-in ceremony.
Like many deaf people, she has some hearing, and can presumably vocalize to an understandable degree, given that her parents didn't even know she was deaf for several years. Thus, it is possible that she took the oath orally.
However, I believe that under the Americans with Disabilities Act, she would be legally entitled to demand that she be allowed to take the oath in ASL, even though ASL is not an official language of the US (though perhaps it ought to be). In particular, state and local governments are subject to the ADA under 42 U.S.C. §§ 12131, and may not exclude individuals with disabilities from "participation in" the activities of said entities (presumably including the mayoral office, though IANAL), by section 12132.