No, U.S. citizens do not need the yellow fever vaccine if they're staying in the U.S. They would normally only need to get a yellow fever shot if they're traveling abroad to areas where yellow fever is endemic, such as parts of South America and Africa.
Although it's common to hear in the media about the presence of the mosquito that spreads yellow fever being in the U.S. (aedes aegypti), actual spread of yellow fever in the U.S. is not a current threat and hasn't been a problem for a long, long time.
In the past, the U.S. has had some severe yellow fever outbreaks, including a very bad outbreak of yellow fever in Philadelphia in the summer of 1793 that led to 5,000 deaths out of 45,000 residents. 15,000 more fled the area. Warmer ports such as New Orleans were especially vulnerable to yellow fever outbreaks that were imported in each summer. Controlling the mosquito population, followed by an effective yellow fever vaccine, is what eradicated the problem in the U.S.