Required disclaimers: I'm not a lawyer I'm not a licensed migration agent I'm am not qualified to answer migration details and must tell you to go see a migration agent.
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During your holiday visa, sure. After that, nope, unless you've got a couple million dollars laying about somewhere for one of the $$$ visas. Otherwise you will have to go for the points based migrations that will give you immediate permanent residency that don't require you to be employed.
Otherwise, if you have a marketable skill and a few years experience, I think it's 5, you could try for a 457 Skilled Temporary Work. The initial visa lasts 3 years usually, but you must maintain employment with an Australian company that is supporting you, but you can switch companies. If you decide to take this route, start looking for a job doing this immediately. The visa could take 6 months to get, could be more, could be less. It's a joyous processes where you totally don't feel like number, and totally not at the mercy of some algorithm applied without mercy.
If she's in an AU Uni, and has very good English, there's a chance she could pass the Skilled Independent as well, and that gives you (both?) more points on the that route.
How many years have you been doing webdev? Got any degrees? Could you move the company here here? Think about these as you look at visas, some may unlock paths.
Something to be aware of, as a working person in Australia. The US tax system is insane. Check out FBAR and know even if you only make money in AU you still have to file US taxes.
Don't even bother looking for ethical banks. Any ethical bank in this country is going to have a No-Americans policy, so you're limited to ANZ, Commonwealth, ING, and a few others. You'll probably want one.
Bring ID, lots of it. http://www.transport.nsw.gov.au/professional-drivers/licences/100-point-check-proving-your-identity You need 100 points for many things, like renting, and opening a bank account, https://www.anz.com/Documents/AU/Commercial/100point_check.pdf