Is there free or dispersed camping in the Shenandoah Valley?
Yes. Dispersed (free, primitive) camping is allowed throughout most of the George Washington & Jefferson National Forest, which wraps both sides of the Valley. Shenandoah National Park is different: overnight backcountry camping there requires a paid permit purchased through Recreation.gov in advance. Dispersed camping is not allowed in Virginia state parks.
National Forest rules
Camp at least 200 feet from water and trails, pack out everything, use established fire rings or a backpacking stove, and stay no more than 14 days in one spot. Some specific areas have additional rules — check the GW & Jefferson NF website before your trip.
Shenandoah National Park backcountry (permit system changed)
Backcountry camping in Shenandoah is no longer free or self-registered. The park replaced paper permits with an online system: permits must be purchased through Recreation.gov (recreation.gov/permits/4675336) or by phone at 877-444-6777, before you arrive. They are no longer issued at entrance stations, visitor centers, or trailhead self-registration boxes. Permits cover up to 14 nights, groups are capped at 10 people (larger groups need a second permit and must camp 50 yards apart), and you may not stay more than 2 nights in one campsite. Campsites must be at least 10 yards from water, 20 yards from any trail or unpaved fire road, and 50 yards from another party. Campfires are prohibited in the backcountry — cook on a backpacking stove. Camping on Old Rag, or hiking the Saddle, Ridge, or Ridge Access trails, also requires an Old Rag day-use ticket. Source: nps.gov/shen backcountry permit and regulations pages.
What you give up
No toilets, no water, no trash service. Bring everything in and out, and have a real waste plan. Dispersed camping is best for experienced backpackers and self-contained vehicle campers.