I thought I will give an update in case somebody else might have similar problems.

I have finally managed to book my flights. The LAN call center was completely useless as well as their email service (I think they have a 30 day reply policy which is ridiculous in our times...).

Anyway, I have found the best contact medium: Facebook! Ha! Their LAN Australia Facebook page is wonderfully helpful. I always got replies very fast, in less than 24h. They acknowledged they had problems with the website and had people working on it and offered to help me book the tickets “offline”. As I was not in a hurry I decided to wait until the website would be fixed. It started working a few weeks ago (about 3 weeks after it stopped working). The booking went smoothly and I got issued the tickets right away.

Just a word of warning, I tried again to make a reservation and pay later, but this part still does not work. However, selecting the flights and booking on the spot works with no problems.
I seriously think LAN should give their Facebook Australia employees a raise!

Good to know Idioteque. I have always had problems with LAN payments. Usually wind up reserving online and paying over the phone.