Budd began her singing career at the age of 12, taking the stage for amateur night at the Tamarack resort camp in the Catskills. There she was spotted by Producer Herb Bernstein, who became her personal manager and arranger. He continued to work with her through the next thirty years. After signing a contract for personal management, he also had her record a demo, signed her to a three-year contract with Metro-Goldwyn-Mayer, and sent her to a voice coach. One of Bernstein's employees suggested her new stage name. Budd appeared on the Merv Griffin Show, The Ed Sullivan Show, The Tonight Show, and The Jim Nabors Hour. In 1968 she released the LP Child of Plenty on the MGM label and became a regular on the NBC TV summer series Showcase '68.

She was born Edith Erdman in Brooklyn, New York, the second of three daughters of Saul and Joan Erdman. Her father was a bottling company executive. Her family was Jewish but not religious. She attended the Roy H. Mann Jr. High School in Brooklyn until 1969, when she transferred to a private academy in Manhattan. In 1993 she contributed her reminiscences of growing up in Brooklyn to the book It Happened in Brooklyn: An oral history of growing up in the borough in the 1940s, 1950s and 1960s (1993).

She performed the title song for the 1972 film Living Free, which played over the opening credits. In 1976 she reached #93 in the charts with the song "One Fine Day", one of her biggest hits. In 1979 she performed on Broadway in They're Playing Our Song. A 1981 New York Magazine article noted that she had "sung in public for 15 of her 27 years", and "almost had the lead in Evita on Broadway".

In acting, Budd is associated with the Circle Repertory Theater and Playwright's Horizons of New York City. She starred in the 1981 Walt Disney film The Devil and Max Devlin as a 19-year-old high school dropout and aspiring singer. Budd has stated that "having the opportunity to work for the Disney Company was a life changer for me. At Disney they always do it right".

In November 1991 she teamed with Burns again to put on a benefit performance to raise funds for the Children's Hospital of Orange County. She performs at the annual fundraiser of the Schultz-Hill Foundation, founded in 2002 in Atlantic City. At the 2004 NYSSBA Annual Convention Preview Convention she performed the songs of Barbra Streisand, with whom she is often compared. On October 22 and 23, 2004, she performed Streisand classics in a tribute program, "The Barbra Streisand Songbook", created by Marvin Hamlisch.

Budd continues to perform mainly in the New York City area and Las Vegas. She released the albums Pure Imagination (1997) and If You Could See Me Now (2000), and to commemorate Sinatra's centenary on December 12, 2015, she has released an album, Remembering Mr. Sinatra. She also teaches master classes in singing.