Azui.com is still down. Looks like the site is having problem with the traffic volume. Since you can't use the UI Weekly Claim System, you can then file your AZUI weekly claim by phone.
Toll Free: 1 (877) 766-8477
Phoenix: (602) 417-3800
Tucson: (520) 884-8477
TTY (Telecommunications Device for the Deaf) 1 (877) 877-6226 Toll-Free