I found this link which shows that they're aren't many hiker/biker sites on the Olympic Peninsula route.
Two things here:
First, that map on the link isn't accurate, and it is missing some hiker/biker state parks. For example, it doesn't list Cape Disappointment, the most SW park on the Pacific Coast. And I know for sure there is a hiker/biker site there. It also doesn't list Beacon Rock in the Columbia Gorge.
Secondly, pretty much all the state parks on the coast are located south of the Olympic Peninsula. On the peninsula itself (at least the west side), there is only one state park between Grays Harbor and Port Angeles: Bogachiel. What you're going to be more concerned with here is the other campground operators: Olympic National Park, the Olympic National Forest, and the other operators. Olympic NP doesn't have hiker/biker sites (at least when I rode it 4 years ago), but park rangers managed to find me a camping spot when we ended up at full campgrounds. None of the spots were great, but serviceable.