Yes, items can make saving throws. But when an item is attended—and a wielded weapon always is—the wielder can substitute their own saving throw bonuses for the item’s, which are usually better. Magic items, and especially artifacts, however, may have superior saving throws than the creature attending them.
Yes, the phrasing “must make a save or” means something unusual happens on a failed save; on a successful save, nothing unusual happens and you can ignore the line entirely and proceed as if it never existed.
Agreed, this is awkward and inconsistent with how “strikes” is typically used within the rules. It seems fair to me to assume that the idea was for the thing actually hitting your skin (whether that’s a sword or an arrow) to shatter, not necessarily the original weapon that the attack was made with.
It’s also unclear whether an arrow in flight is “attended” by the creature who shot it, and thus can use that creature’s saving throws. Pretty sure you’ll just have to ask your DM, because I don’t think the rules ever touch on it.
But ultimately, Vow of Peace is an awful, awful feat. Beyond the awkward mechanics and open rules questions discussed above, it ruins the game, not just for you, but for everyone playing. If you want to play a game that focuses on non-violent approaches to dealing with problems, please, get a better system for that than D&D. D&D has only extremely basic non-combat rules, and what rules it has will generally impede, rather than improve, your game. But if you insist, make absolutely certain to get the approval and buy-in from everyone in the game—not just the DM, everyone—before you show up with a character who has this feat. Completely warping the nature of the game played as a unilateral decision made without consulting the other players in the group is rude in the extreme. I have seen people kicked out of groups for taking Vow of Peace, and I have left groups where someone took it and wasn’t kicked out.
And, for the record, Vow of Peace is the worst case of this in all of 3.5—but really, the whole of Book of Exalted Deeds is questionable in the extreme. There are a few decent things in there, but between the bizarre moralizing by WotC authors, the blatant contradictions of other sources on various subjects of alignment, and the book’s propensity for declaring “X is evil; here use Y instead, it’s exactly like X except not evil for some reason,” the whole book should be viewed with extreme skepticism in my opinion. The last gets decidedly creepy when you see that one of the Xs for which it provides a Y is mindrape.
