There are a number of places to rent kayaks in the harbor at Avalon, but Descanso Beach Club (just past the old casino building) happily rents kayaks overnight or longer for paddling trips along the islands' coast. My kayak was waiting for me when I arrived for my overnight trip. After loading my gear into the fore and aft storage hatches, one of the club's managers sat down with me and my map and gave me detailed directions/instructions on what to look for and where to go to reach my campsite.
Avalon and most of the beach campsites are on the lee side of the island, but the winds can still cause some decent swells in the early afternoon. The campsites are listed as primitive, and they really are, just a rocky beach barely wide enough to pitch your tent. You must bring your own water and toilet bag, which is required. You will see lots of private fishing boats along the way. Although Avalon is popular during the summer and on weekends, most of the primitive sites are probably readily available except for holidays.