Applications to the legacy Carriers only ask if you have the legal right to live and work in the US. Now as far as I'm aware, this E3 visa allows you to apply for Permanent residency, i.e., a Green Card. I don't know at which point you can apply (the H1B visa allows you to apply for your GC as soon as you get the visa) but you used to be issued a temporary residency permit until your GC is issued, which covers the legal right to live and work part of an application.
In other words...yes. As long as you have or are applying for a green card.