NCR I don't think would waste their ammo on it, if they would, they would have likely fired more than once until it fell to the ground, realisticly ED-E wasn't going fast enough to evade any bullets so they'd have destroyed the bot.
Enclave have more important things to do, like survive and integrate into the Wastes. They would shoot their own bot who was sent across the country to one of their own (ex)bases, another note is that they are highly advanced people dealing with energy weapons so a .308 round is not going to be used by them, maybe .50 cal but not a .308, its just too primitive for them, also they wouldn't hire someone to take it down either because that would almost endanger their integration efforts and the said hitman would have no mode of transport that could keep up with ED-E, nor would s/he have the tech good enough to track ED-E across the country. Hitmen need to eat and sleep v ED-E who need neither.
....After looking at the ammo used, and referring to the wiki which says it was a sniper bullet, .308 round, and what factions use .308 weapons only 3 exist: NCR, Legion and Fiends - ignoring some NPCs. As for it being a "sniper shot" I would disregard it being from an actual Sniper Rifle but instead from a Hunting Rifle, possibly scoped - Boone
, since only NCR and few Legionnaires have Sniper Rifles.....Forgot where I was going with this. - Khans wouldn't shoot it for lolz either, Fiends would.
It was Fiends, with a Hunting Rifle, in the Mojave Desert.