http://www.dgif.virginia.gov/hunting/regulations/furbearerhunting.asp#coyote Look under fur bearers, then for coyotes.
Coyote
Season:
Continuous open season on private lands. Coyote hunting on National Forest Lands and Department Lands is permitted from September 1 through March 10 and during the spring turkey season and spring squirrel season on certain areas (see area listings for spring squirrel season). Coyotes may not be hunted on Sundays.
Coyote Bounty Law
Coyote bounties are not administered by DGIF. Counties have the option of establishing their own coyote bounty system. For more information, hunters and trappers should contact their County Administrator or County Board of Supervisors.
Seems the fibbie guy has you over the proverbial barrel of actual fact.
There is no "varmint disposal" method of taking coyotes - per the DGIF ya gotta hunt them. You also must have a license to hunt them.
http://www.dgif.virginia.gov/hunting/regulations/licenses.asp You can buy and print them off on-line - check the sidebar.
Remember you gotta have a license to hunt the crow you are gonna be eating. And you can hunt them Monday, Wednesday, Friday, and Saturday only. Apparently Tuesdays and Thursdays they play golf with the doctors.
http://www.dgif.virginia.gov/hunting/regulations/smallgame.asp#crow Follow the regulations or you are going to have a very unpleasant talk with Mr. Game Warden.
stay safe.