Mr Shaw collapsed and died on the platform of Margate railway station in Kent on Friday evening, according to his friend, the author and royal biographer Michael Thornton.
In 50 years in the showbiz, Mr Shaw managed a diverse range of clients from Booth, Prime Minister Tony Blair's father-in-law to Margaret, Duchess of Argyll.
Mr Thornton said he was shocked at the death of his friend, regarded as a powerful figure in the West End and one of theatre's most colourful characters.
"Vincent had had health problems during the last two years, but nothing that was regarded as life-threatening," he said.
An inquest is expected to be held, he added.
Other leading figures from the West End paid tribute to Mr Shaw, who lived near the Royal Albert Hall in west London.
'Great man'
The impresario Bill Kenwright said: "This is truly the end of an era. He was the last of the old-style West End agents.
"He was also a great man. I began my own career in management from a small desk in the corner of his office, and any success I have had owes a huge amount to his help and friendship."
Mr Shaw's former wife, actress Lisa Rayne, said she felt "quite bereft".
The couple married in 1953 and divorced three years later, but remained friends and he continued to manage her.
Born Ronald George Shaw on 14 May 1925 in Westcliff-on-Sea, Essex, he was educated at Stationers School.
His first marriage at the age of 17 to Gladys Skull ended in divorce three years later, with Shaw recalling: "It must have been her name that put me off."
He entered showbusiness at 18 after serving a brief prison sentence for refusing to enlist for military service in the World War II.
Lucrative move
After a short spell in acting, he switched to directing and helped pioneer the guest-star tradition in British repertory theatres.
The move to management proved lucrative, making him a millionaire by the 1960s.
Although he set many actors on the road to success, his biggest box-office success was Sooty's Christmas Show, which he presented annually for 30 years.
He turned the bear into a money-spinning industry, negotiating substantial contracts with Thames Television and taking the stage show across the country.
'Beautiful women'
At the time of his death, Shaw was working on a candid autobiography, which he intended to call This Way to the Pit.
Mr Thornton said it was a shame his friend had not finished the book.
He added: "There were so many women in his life over the years that I lost count of them.
"He was a man who simply loved beautiful women and most of them loved him in return and remained fond of him afterwards."
Mr Shaw had no children by either of his marriages.