Legally you need permission from the State of California to install a mooring in the waters of California-anywhere with in three miles of land. However there are plenty of moorings for rent on a first come first served basis on Catalina. All the moorings are owned and operated by either the city of Avalon or Two Harbor Enterprises. You can anchor for free any where on the island so long as you are at least 100 yards from any established mooring. There are a couple exceptions but the harbor patrol will let you know if you are in the wrong place.
With the exception of Cat Harbor most of the decent anchorages on the west end of Catalina have moorings in them operated by Two Harbors. They can be reached of VHF ch 9.
If you have any specific question just ask I've lived aboard on the island for 5 years and used to work for Isthmus harbor patrol.