No.  Visitors are not permitted to collect any parts of migratory birds (including nests and feathers) or parts/bones of whales, dolphins and sea turtles.  These actions are prohibited by several laws that apply throughout the United States and not just on Virginia’s seaside:
The collection of all bird feathers, nests, and bones is strictly prohibited under the Migratory Bird Treaty Act.  This statute makes it unlawful to pursue, hunt, take, capture, kill, or sell birds listed as migratory birds and also grants full protection to any bird parts including feathers, eggs, nests, and bones.
The collection of bones or any parts from any type of marine mammal (dolphins, whales, seals, etc.) is strictly prohibited under the Marine Mammal Protection Act.  This act prohibits the “take” of marine mammals in U.S. waters and the importation of marine mammals and marine mammal products into the U.S.
The collection of any sea turtle bones, shells, or other parts is strictly prohibited by the Endangered Species Act.  All five species of sea turtles are listed as federally threatened or endangered and therefore, any action that causes “taking” of these species is prohibited.