You can legally paddle this coast privately. Most people should not, and the rules are stricter than they look.
Hawaii's Department of Land and Natural Resources sets the season for this coast: paddling it is permitted May 15 through September 7, and that is also the window in which kayaks may land — at exactly two places: Kalalau Beach, with a valid camping permit, and Miloli'i Beach, for camping with a permit or for day use. Landings are prohibited at every other beach in the park, Hanakapi'ai, Honopu and Nu'alolo Kai included. Camping along the coast without a valid permit can get your kayak confiscated — including a rental.
Commercial operators need a state permit to land passengers here at all. DLNR publishes the list, and there are three kayak companies on it: Kayak Kauai, Na Pali Kayak and Outfitters Kauai, for guided landings at Miloli'i, May through September. If a company is not on that list, ask why.