I’m surprised that this question has been asked as a search term in Google. I can answer the question without referring to any books or any Internet websites. I don’t need to look up any law. The answer is plainly obvious; although respondents on one pet forum failed to come up with an answer.
If a person shoots a cat with a BB gun the cat is likely to be injured and feel pain. A BB gun is a form of airgun which shoots pellets at high-speed. If the person who shot the cat lives in a country where there are animal welfare laws, the laws will invariably be concerned with this sort of behaviour. Animal welfare laws cover various circumstances but one of the prime circumstances for which they are created is to protect animals from violent abuse by people. The abuse can take various forms. A very typical form is to shoot at cats either with BB guns, air rifles or rifles shooting bullets. Animal welfare laws are criminal statutes: they create crimes.
Therefore, it is 99% certain that it would be illegal (and criminal) to shoot a cat with a BB gun because it would be causing unnecessary suffering to the animal. I can’t think of any circumstances which would justify a person shooting a cat with a BB gun. Shooting a cat with a .22 rifle, for example, might on very rare occasions (rabies or harassing livestock?) be justified. Also if a bullet shot was to the head at close range and the cat died instantly in a form of euthanasia it may be acceptable in some states but even under these circumstances the cat would experience severe pain and suffering albeit for a split second I would argue.
Cat haters will say that they are allowed to shoot cats but they get the law wrong all the time (note: if cat haters comment they must present the law accurately. I won’t accept rants and deviant opinions from nutters).
A BB gun will not kill a cat unless the cat is shot many times and the pellets hit a sensitive part of the body. Therefore the cat is bound to feel pain and suffer needlessly. The usual reason why young men (your typical miscreant) shoot cats with BB guns is for the fun of it. This is not a justifiable reason in law. Therefore don’t do it. You will be a criminal. You might well get away with it because rarely are there criminal prosecutions against people shooting cats with BB guns but don’t bank on it.
One reason why cat abusers get away with it is because there will often be a lack of evidence – another reason to keep cats inside. BB guns are mainly an American product while in the UK there are air guns – same difference. They are often used to shoot at cats. Wandering outside cats are excellent, live, moving targets for the imbeciles who think it is fun to do this.