A bb gun can kill a frog, but you would have to hit it perfectly in the brain, i would bet smaller than a pea if you even figure out exactly where it is at. I have seen them hop off after a solid hit with a daisy 880. An airgun will work, but I use a .25 cal magnum breakbarrel. Probably overkill, it makes a mess, but all you ever eat are the legs, so it works.
I usually go for a headshot still. Use an airgun with good power and shoot in the "head".