I doubt you will be able to join a US library without paying a fee, though as jhowell and others have pointed out, there are US libraries which will accept you if you are prepared to pay. Such libraries will usually have a much wider variety than in Australia and earlier access to new releases.
Within Australia, an online search and a quick enquiry will find many libraries that you can join in your own state. I am a member of a number of libraries in different areas of the City I live in. Some you can even join online, though most seem to require a visit and some identification. Unfortunately ebooks for Kindle seem to be available from libraries only in the US. Most Australian libraries seem to use Overdrive for epubs and pdf's.
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