It's a common misconception that Weeping Angels are made of stone. They are made of stone, but only when you look at them. It's a natural defense mechanism, because as The Doctor said himself "you can't kill a stone". The full quote if you're wondering is this "In the sight of any living thing, they literally turn to stone. And you can't kill a stone."
This means that when you aren't looking at a Weeping Angel, they're a living breathing creature made of meat and bone. That's why they can move when they're not being observed. It's because, when they're not being observed, they're not stone.
Of course, whether or not they can be killed by shooting at them with your eyes closed, is a completely different matter. Some would argue that a Weeping Angel would be able to dodge a bullet, but a bullet travels at 954 Miles Per Hour, and even if a Weeping Angel could move at half that speed (which would be fast enough to dodge the bullet in my opinion), why does it take so long for a Weeping Angel to reach you whenever you blink? Clearly a Weeping Angel doesn't move that fast. But that isn't the reason why I don't think shooting at them will work. The reason why I don't think shooting at them will work is because we've seen them regenerate from practically nothing.
Chances are, shooting at them with your eyes close will only slow them down, not kill them.
I've talked about this with someone called Vindurza, for quite a while. I was constantly referencing the Tenth Doctor, and he was constantly referencing the Eleventh Doctor. We didn't come to an agreement. So as you can see, this is a highly debatable question. What are your thoughts?
It makes you wonder why they even need to turn to stone anyway.