Upvote:4
You will have absolutely no issues. By far the majority of Sydney and Melbourne petrol stations have LPG, especially the chains, so assuming your car has 400 kilometres range (600 km was not unreasonable five years ago) on a full tank minimum, you can make the following stops without even going out of your way:
Upvote:5
You should be fine - people have done it in the past, even driving Melbourne to Gold Coast, with plenty of stops.
However, if you want to be certain, the Shell gas stations have an online Fuel finder site to locate the different types of stations, as well as iPhone and Android apps. And they're certainly not the only fuel company in Australia.