The details of imbue arrow are... ambiguous. It gives you an extra special way to cast a spell, but offers no details on which spells are allowed (beyond that it be an “area” spell, which is presumably one that has an Area entry but even that might be argued). As written, you could argue that you can pick any area spell ever, regardless of whether or not you are actually able to cast it yourself. And that when you cast it this way, it uses no resources, because you’re casting it via your imbue arrow ability, which makes no mention of resources, rather than through, say, a “spells” ability, which does explain how your limited resources work.
Needless to say, that’s pretty unlikely to be what they meant. Unfortunately, that leaves us in a bit of quandary about what they did mean.1 At a minimum, anyway, I am going to assume that you have to pick some spell you otherwise can cast, and that when you imbue an arrow with it you use up resources as if you had cast it “normally.”
Based on the use of the word “spell,” I am also going to say spell-like and supernatural abilities are out. Despite being “like” spells, spell-like abilities are not spells and things that references spells generally mean spells and not spell-like abilities. See this Q&A for somewhat more discussion on this. And supernatural abilities are even more removed.
A spell-storing weapon is also probably out—you can only access the spell in a spell-storing weapon by attacking with the weapon, and since it is giving you a special way to cast that spell while imbue arrow wants you to use its special way of casting the spell, they wouldn’t be compatible.
That leaves wands, scrolls, or a ring of spell storing. And happily, all three of these allow you to “cast” spells:
A minor ring of spell storing contains up to three levels of spells that the wearer can cast.
(Ring, Spell Storing, Minor)
If she meets all three requirements but her own caster level is lower than the scroll spell’s caster level, then she has to make a caster level check (DC = scroll’s caster level + 1) to cast the spell successfully. [...]
A spell successfully activated from a scroll works exactly like a spell prepared and cast the normal way.
(Scrolls, Activate the Spell and Determine Effect)
Wands use the spell trigger activation method, so casting a spell from a wand is usually a standard action that doesn’t provoke attacks of opportunity.
(Wand, Activation—note that Staff, Activation says the same thing about staves)
There is still room to argue that these don’t work, however: each of these requires that the user activate the item in order to cast the spell, and imbue arrow doesn’t say anything about activating anything as part of its special casting ability. And again, since imbue arrow doesn’t really explain itself very well, we’re already kind of making up its requirements here.
So ultimately, while we can parse the rules here, we cannot eliminate the ambiguity. You’re better off just making a ruling yourself, as DM. It certainly sounds like you’re OK with this. I would be too—arcane archer is a very weak class, and imbue arrow is about the only thing going for it. And since this rogue likely will not have the ability to use antimagic field—the only thing you can imbue an arrow with and make arcane archer actually worth it—I see little danger in allowing it. And for that matter, I’d allow spell-like and supernatural abilities too—that’s much less ambiguous, but I still see no reason to deny it. And using magic items is a better usage of arcane archer than most I’ve seen.
- It’s worth noting that there are other abilities that similarly involve casting a spell with a weapon. They are generally written more tightly, preventing these ambiguities. This Q&A discusses through several of them, for example—including a spell-storing weapon.