Certainly you can paddle from Avalon to beach camping. Willow Cove and Long Pt. Beach would be the closest primitive camp spots and are doable in a days paddle. Then working your way to 2 harbors you have Italian Gardens, Goat Harber, Gibraltar Cabrillo, Lava Wall, Paradise, Rippers in order.
You may want to contact Descanso Beach Ocean Sports to see how much they would charge to allow you to lock up your kayak on their rental property. Then you would just get in and go after walking there from the boat.
http://kayakcatalinaisland.com/
You could also ask Wet Spot Rentals
http://www.catalinakayaks.com/index.php/location/
Storing at these locations you don't have to worry about cabs or trailers.