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 requires a backcountry permit (free) for overnight stays; no dispersed camping is 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
Free backcountry permits are required and can be picked up at park entrance stations or self-registered at trailheads. Camping is allowed in most of the park's backcountry with setback rules from trails, water, and roads.
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.