The CASA version is actually hosted at aviationidaustralia.net.au
(it even says so if you click 'yes' to the Have you ever been issued an ASIC by CASA? question.)
I tried it using Seamonkey (Mozilla) and it worked fine.
Out of idle curiosity I tried doing a 'normal' AviationID application, and apart from the branding logo at the top of the page it appears to be identical to the CASA one.
It's all being done by https:// so your data-entry and C/C details should be ok.
I was amused by the year-range allowed in the mandatory 'Residence Since' field. (honest guv, I have been living at the above address since 1908)