Based on other similar questions, I've determined that a familiar can attune to magic items. But can a familiar cast spells from the Ring of Spell Storing? The description of Find Familiar states "A familiar can't attack, but it can take other actions as normal." Does this mean if it could cast spells, it could only cast non-attacking spells?
Yes, depending on how the spell targets creatures
Rubiksmoose gave an excellent break down of what an attack is here, but what's relevant to this question can be found on p.193-194, PHB:
Whether you're striking with a melee weapon, firing a weapon at range, or making an attack roll as part of a spell, an attack has a simple structure
If there’s ever any question whether something you’re doing counts as an attack, the rule is simple: if you’re making an attack roll, you’re making an attack.
As a familiar is not able to attack (excluding a Warlock Pact of the Chain familiar), but there is no mention to a familiar's ability to cast a spell, then the familiar should be able to cast any spell that does not require an attack roll.
The other question that may come up is the familiars ability to complete the components of the spell, but as answered here in regards to magic items casting spells, DMG p.141 states:
The spell is cast at the lowest possible spell level, doesn't expend any of the user's spell slots, and requires no components, unless the item's description says otherwise.
So the spell would not require components. In regards to the Ring of spell storing, provided the spell used from the ring does not require an attack roll, then it is perfectly acceptable for the familiar to use the ring.