All campgrounds in Olympic National Park (NP) are "first-come-first-serve", except for Kalaloch (pronounced "clay-lock"):
Campgrounds - Olympic National Park (U.S. National Park Service)
We stayed at several campsites in Olympic NP this past spring; and while all were beautiful, we especially enjoyed the campground and activities at the Hoh (River) Rain Forest. See photos in link below:
You may also want to stay at least one night at the Washington Land Yacht Harbor (Airstream) RV Park in Olympia, WA. With a Costco and WalMart within a couple of blocks and easy access from I-5, it makes a great place to rest and restock either on the way up or on the way back to Arkansas (assuming that's your roadtrip origination point).
Lots to see and do in Seattle/Tacoma area. Don't miss the Museum of Flight and Boeing Tour. And, if you haven't seen enough planes by then, also stop in McMinnville, Oregon to see the Spruce Goose (and lots more) at the Evergreen Aviation & Space Museum.
Note: If you go to either the Museum of Flight or Evergreen Museum, buying an annual membership will get you free or discounted admission to both, plus about 300 other science and technology museums world-wide. For more info, see below:
Flight Museum Membership Advantages Include Aloft Magazine, Science Centers, Free Events, etc. | The Museum of Flight
Send me a PM if you'd like more information on our roadtrip itinerary in the northwest.