Premier Gladys Berejiklian announced the "exciting" development at the opening of the government-run Homebush hub on Monday. Visitors to aged-care facilities are still limited to two people per day.
From Wednesday, anyone 50 or older can register to get the AstraZeneca vaccine at the hub from May 24.
"If you don't have a GP or don't have access to a GP, or you find it hard getting to them, you can book with us from 12 May and get your jab from 24 May", she said.
Sydney's snap COVID-19 restrictions introduced on Thursday have been extended for another week, with one significant change to mask rules.
"Despite extensive investigations to date, NSW Health has not identified how the initial case, the man in his 50s, was exposed to COVID-19, which suggests he acquired the infection through brief contact with a now unidentified person who was infectious in the community", NSW Health wrote in a statement.
Asked about the identity of the so-called "BBQ man", Chief Health Officer Dr Kerry Chant said she was "incredibly disappointed" to see him being named in the media.
Genomic testing has linked the man's case to that of a returned traveller from the USA but how the virus was transmitted between them has stumped authorities.
All close contacts of the couple have returned negative results, with Dr Chant believing contact with the mystery infection would have been "very fleeting".
"We're still concerned that there may be chains of transmission in the community that are yet unrecognised", she said.
It will operate six days a week from 8am to 8pm in a state-of-the-art specially fitted out building.
She said that was not in accordance with the wishes of the gentleman.
NSW Health Minister Brad Hazzard also weighed in on the matter, saying it was "absolutely appalling".
"It's been a war ever since".
The continuing restrictions mean that for the next week household gatherings remain capped at 20 people, masks are compulsory on public transport and in indoor venues such as theatres and aged care homes, and singing and dancing remains mostly banned.
Hospitality patrons are still not permitted to drink while standing, although retail customers are no longer obliged to wear masks.