Camp at the very popular Twin Harbors State Park (two miles south of town, 888/226-7688, www.parks.wa.gov [1], $17 tent, $21 full hookups, $7 extra reservation fee). Some campsites are just steps away from the beach, and showers are available. The campground is open late February–October.
Westport [2] is jam-packed with RV parks catering to the fishing crowd. Perhaps the most laden with amenities, American Sunset Resort (360/268-0207 or 800/569-2267, www.americansunsetrv.com [3], $29 full hookups, $19 tent) offers a heated outdoor pool, horseshoe pits and playgrounds, showers, and laundry facilities, plus free Wi-Fi. The property is pet friendly, offers plenty of space for larger rigs, and sports a convenience store and a special area for tent campers. It even rents out two permanent trailers parked at the property.
Links:
[1] http://www.parks.wa.gov
[2] http://www.moon.com/destinations/washington/olympic-peninsula-and-the-coast/grays-harbor/westport
[3] http://www.americansunsetrv.com