This might help. Here are the statute's pertaining to the disposition of shelter animals:
"§ 3.2-6548 Animal shelters; confinement and disposition of animals; affiliation with foster care providers; penalties; injunctive relief.
"A. An animal shelter may confine and dispose of animals in accordance with the provisions of subsections B through G of § 3.2-6546."
And,
"§ 3.2-6549 Releasing agencies other than pounds or animal shelters; confinement and disposition of companion animals; recordkeeping; affiliation with foster care providers; penalties."
(I'm posting the sections that pertain directly to this discussion, but you can look at the material in its entirety here: https://vacode.org/3.2/65/)
D. If an animal confined pursuant to this section has not been claimed upon expiration of the appropriate holding period as provided by subsection C, it shall be deemed abandoned and become the property of the pound.
Such animal may be euthanized in accordance with the methods approved by the State Veterinarian or disposed of by the methods set forth in subdivisions 1 through 5. No pound shall release more than two animals or a family of animals during any 30-day period to any one person under subdivisions 2, 3, or 4. 1. Release to any humane society, animal shelter, or other releasing agency within the Commonwealth, provided that each humane society, animal shelter, or other releasing agency obtains a signed statement from each of its directors, operators, staff, or animal caregivers specifying that each individual has never been convicted of animal cruelty, neglect, or abandonment and updates such statements as changes occur; 2. Adoption by a resident of the county or city where the pound is operated and who will pay the required license fee, if any, on such animal, provided that such resident has read and signed a statement specifying that he has never been convicted of animal cruelty, neglect, or abandonment; 3. Adoption by a resident of an adjacent political subdivision of the Commonwealth, if the resident has read and signed a statement specifying that he has never been convicted of animal cruelty, neglect, or abandonment; 4. Adoption by any other person, provided that such person has read and signed a statement specifying that he has never been convicted of animal cruelty, neglect, or abandonment, and provided that no dog or cat may be adopted by any person who is not a resident of the county or city where the pound is operated, or of an adjacent political subdivision, unless the dog or cat is first sterilized, and the pound may require that the sterilization be done at the expense of the person adopting the dog or cat; or 5. Release for the purposes of adoption or euthanasia only, to an animal shelter, or any other releasing agency located in and lawfully operating under the laws of another state, provided that such animal shelter, or other releasing agency: (i) maintains records that would comply with § 3.2-6557; (ii) requires that adopted dogs and cats be sterilized; (iii) obtains a signed statement from each of its directors, operators, staff, and animal caregivers specifying that each individual has never been convicted of animal cruelty, neglect, or abandonment, and updates such statement as changes occur; and (iv) has provided to the pound, animal shelter, or other releasing agency within the Commonwealth a statement signed by an authorized representative specifying the entity’s compliance with clauses (i) through (iii), and the provisions of adequate care and performance of humane euthanasia, as necessary in accordance with the provisions of this chapter. For purposes of recordkeeping, release of an animal by a pound to a pound, animal shelter or other releasing agency shall be considered a transfer and not an adoption. If the animal is not first sterilized, the responsibility for sterilizing the animal transfers to the receiving entity. Any proceeds deriving from the gift, sale, or delivery of such animals shall be paid directly to the treasurer of the locality. Any proceeds deriving from the gift, sale, or delivery of such animals by an animal shelter or other releasing agency shall be paid directly to the clerk or treasurer of the animal shelter or other releasing agency for the expenses of the society and expenses incident to any agreement concerning the disposing of such animal. No part of the proceeds shall accrue to any individual except for the aforementioned purposes. E. Nothing in this section shall prohibit the immediate euthanasia of a critically injured, critically ill, or unweaned animal for humane purposes. Any animal euthanized pursuant to the provisions of this chapter shall be euthanized by one of the methods prescribed or approved by the State Veterinarian. F. Nothing in this section shall prohibit the immediate euthanasia or disposal by the methods listed in subdivisions 1 through 5 of subsection D of an animal that has been released to a pound, animal shelter, other releasing agency, or animal control officer by the animal’s rightful owner after the rightful owner has read and signed a statement: (i) surrendering all property rights in such animal; (ii) stating that no other person has a right of property in the animal; and (iii) acknowledging that the animal may be immediately euthanized or disposed of in accordance with subdivisions 1 through 5 of subsection D. G. Nothing in this section shall prohibit any feral dog or feral cat not bearing a collar, tag, tattoo, or other form of identification that, based on the written statement of a disinterested person, exhibits behavior that poses a risk of physical injury to any person confining the animal, from being euthanized after being kept for a period of not less than three days, at least one of which shall be a full business day, such period to commence on the day the animal is initially confined in the facility, unless sooner claimed by the rightful owner. The statement of the disinterested person shall be kept with the animal as required by § 3.2-6557. For purposes of this subsection, a disinterested person shall not include a person releasing or reporting the animal.