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A Reserve Ioun Stone can store up to three levels worth of spells and can be used by anyone.
While the spell uses the same stats as the original caster (DC, spell attack, and ability modifier, slot used if upcast) the follow up text says that the spell is treated as if cast by you.
Does this mean I could hire a wizard to cast Find Familiar three times into the stone and pass it around the party, giving everyone a free familiar?
Unless a DM rules that there is a limit, casting Find Familiar from the Reserve Ioun stone provides a familiar per the spell description.
Any creature can cast a spell of 1st through 3rd level into the stone by touching it as the spell is cast. (SRD p. 227)
Your PC's are creatures and thus qualify under "any creature."
The spell description for Find Familiar (SRD p. 149) does not impose any limit on class or race. With no specific prohibition, there's no reason not to allow it beyond a DM's preference for a particular world.
A hard requirement that you didn't address in your question is that each character in turn needs to attune to the stone in order to cast the find familiar spell. (SRD p.227). That will take at least three short rests to accomplish.
Ioun Stone Wondrous item, rarity varies (requires attunement)
How to attune the stone: one short rest per player character
Attuning to an item requires a creature to spend a short rest focused on only that item while being in physical contact with it ... at the end of the short rest, the creature gains an intuitive understanding of how to activate any magical properties of the item, including any necessary command words. An item can be attuned to only one creature at a time (SRD p. 206)
