Lots of folks are under the impression that there are only 2 or three types of firearms that could fire this round. In fact there are actually single shot handguns that could easisly (yes, easily) accomplish the kind of shooting being done.
Classic military sniping is usually done with heavier calibers, but what's going on here isn't classic military sniping in any sense of the word.
But, to answer what I think was the gist of your question, the .5.56mm round isn't usually chosen by persons who would consider themselves a military sniper. When I attended Quantico, I trained with a bolt action Remington chambered for the 7.62mm x 51 round. (That's also sold commercially as caliber .308) I was also trained to do long range shooting with an accurized M-14. That weapon also chambers the 7.62 x 51 cartridge.
Now, to address what I think was the second part of what you wanted to know, the 5.56mm (also sold commercially a caliber .223) is capable of remarkable accuracy at ranges well beyond 300 yards. For a while, .223 was a favorite among people who hunt prairie dogs routinely at ranges beyond 200 yards. FYI, a prairie dog is about the size of your average house cat.
Sorry to get long winded on you, but I'm getting real tired of people calling whoever is killing these people a 'sniper'. I was a 'sniper', and quite frankly hanging that moniker on the chunk or chunks of human debris who are committing these murders insults me.
L