I would love to know this too as I am in Sydney as well. If I ever find any, will let you know. I think there are a few 'organic' hair salons .... not sure if the products are vegan though .... what part of Sydney are you?
I'm in the Shire .... yes, I've gone to one salon that uses Aveda in Cronulla. I sort of like their products but to me they're not perfect as they still have a few ingredients I don't like such as parabens, so they're not as great as they claim to be in my book and charge quite a bit for it ... though I guess at least they are better than the ones that don't even try.
There is one that I drive by on the way to work that says 'organic' in big writing on it's window ... it's on southern cross drive under the new apartments along the ED. They say they are open for late nights too but have no idea what products they use. If I manage to grab their name one morning I'll let you know.