I'm sure it's fairly easy to look up Richmond and other surrounding counties. I happened to have the relevant sections bookmarked for Henrico and Hanover.

Henrico: https://library.municode.com/va/henrico_county/codes/code_of_ordinances?nodeId=CD_ORD_CH24ZOOR_ART4USRE_DIV4ACUSST_S24-4419ACKEANOUDWACDW

Poultry.

The keeping of poultry is allowed as an accessory use to a single-family detached, duplex, or manufactured home dwelling in accordance with the standards in this subsection (see Sec. 24-4302.A, All Animal Husbandry Uses, for standards for keeping poultry as a principal use).

1. The keeping of poultry is allowed by right only in the A-1, Agricultural District. In the R-0, R-0A, R-1, R-1A, R-2, R-2A, R-3, R-3A, R-4, and R-4A districts, the keeping of poultry accessory to a dwelling is allowed subject to approval of a Conditional Use Permit in accordance with Sec. 24-2308, Conditional Use Permit.
2. No more than six hens may be kept on the property. Roosters are prohibited.
3. The hens must be kept within a covered enclosure that includes a coop and an enclosed run. The coop and the enclosed run must have a combined minimum enclosed area of 30 square feet. The enclosure must be located in the rear yard only and must meet a side yard setback of 25 feet and the rear yard setback that applies to the dwelling.

Hanover: https://library.municode.com/va/hanover_county/codes/code_of_ordinances?nodeId=COCO_CH26ZOOR_ART5SIDERE_DIV8RESPUS_S26-298.1KECHREDI

Section 26-298.1. - Keeping of chickens in residential districts.

Notwithstanding any district regulations to the contrary, the keeping of chickens on lots with single-family detached homes within the RS, R-1, R-2, and R-3 districts shall be permitted as an accessory use so long as the following conditions are satisfied:

1. There shall no more than six (6) female chickens (hens) kept on the property. No male chickens (roosters) are permitted.
2. All chickens shall be kept within an enclosure on the property. This enclosure shall:
   a. Contain a coop to house the chickens;
   b. Be constructed entirely within the rear yard of the property; and
   c. Be a minimum of fifteen (15) feet from all side and rear lot lines.
3. All areas associated with the keeping of chickens shall be cleaned and made free of waste on a regular basis. Waste associated with the use shall be disposed of in an appropriate waste disposal container that is periodically removed from the site.
4. The property owner shall employ a means of eliminating any odor problems and propagation of insects related to the use.