Carr died in Los Angeles after complications from a rare form of dementia, her representative said.
In her role in the 1965 film, she famously performed the song Sixteen Going on Seventeen.
After leaving the film industry, Carr ran an interior design firm in California.
Her mother had arranged for her to audition for the role, although she had had no singing or acting lessons.
Rodgers and Hammerstein’s The Sound of Music was a massive hit, at the time surpassing Gone with the Wind as the highest-grossing film of all time.