Not true. The only person allowed to legally shoot a dog chasing deer is a conservation officer and then only after attempting to catch it.

From the WV Code:

§20-2-16. Dogs chasing deer.
No person shall permit his dog to hunt or chase deer. A conservation officer shall take into possession any dog known to have hunted or chased deer and the director shall advertise that such dog is in his possession, giving a description of the dog and stating the circumstances under which it was taken. Such notice shall be published as a Class I legal advertisement in compliance with the provisions of article three, chapter fifty-nine of this code, and the publication area for such publication shall be the county. He shall hold the dog for a period of ten days after the date of the publication. If, within ten days, the owner does not claim the dog, the director shall destroy it. In this event the cost of keeping and advertising shall be paid by the director. If, within ten days, the owner claims the dog, he may repossess it on the payment of costs of advertising and the cost of keep, not exceeding fifty cents per day. A conservation officer, or any officer or employee of the director authorized to enforce the provisions of this section, after a bona fide but unsuccessful effort to capture dogs detected chasing or pursuing deer, may kill such dogs.

Now granted that may be one of those laws that aren't enforced very much. But I can assure you if you shoot someone's dog, whether it was chasing deer or not, and the owner finds out and decides to pursue legal action, you my friend will be in a lot of trouble.

As others have said on here, there are legal ways of handling it. Best to attempt those first. Remember CYA.