The 52-year-old, who had been performing in the show, The Secret River, died on Sunday.
She was a regular at the festival but was best known for acclaimed performances in films including Rabbit-Proof Fence. Sydney Theatre Company said they were "absolutely devastated" by the news.
Two shows in Edinburgh were cancelled last week after Ms Lawford-Wolf fell ill before one of her cast mates stepped into her role as narrator for the remaining four performances.
'Incredibly talented' In a statement Sydney Theatre Company paid tribute to the actress. It read: "Ningali was an incredibly talented performer as well as a wonderfully caring and thoughtful person. We've lost one of Australian theatre's greatest treasures."
Ningali was a Wangkatjungka woman born under a tree at Christmas Creek Station in the far north Kimberley region of Western Australia.
She trained as a dancer at the Aboriginal Islander Dance Theatre, then continued to perform professionally with Bangarra Dance Theatre in Sydney to build a stage and screen career.
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