You can apply class abilities to spell casting from magical items
While casting from an item is not the same as a Casting a Spell action, it is still casting a spell. Unless a term is given a special definition in 5e game terms, you use the common definition for that term. You are casting the spell, the ability to do so comes from the item. For instance, the Staff of Fireballs:
.. you can use an action to ... cast one of the following Spells ...
Is no different, in this regard, to Wand of Fireballs from:
you can use an action to ... cast the Fireball spell ...
Jeremy Crawford has posted a number of answers we can point to in the time since this question appeared that verifies this really is the case. Each time he's been asked about a specific class ability the answer has been an affirmative. These were made when his tweets were official rulings, but since have been "downgraded" by the Sage Advice Compendium to his personal advice and perspective. While this is true, it still is clarification of the intent of the designers.
Metamagic
Can Metamagic be used on magic item casting a spell? Yes.
If a magic item's description says you cast a spell from it, you can use Metamagic on the spell. #DnD
and
Metamagic works w/ any spells that sorcerers cast. Wild Magic Surge can work w/ any sorcerer spell they cast. #DnD
Wild Magic
Can using a magic item cause a Wild Magic Surge? Yes
Yes.
Arcane Ward
Can casting from a wand/staff recharge Arcane Ward? Yes
Arcane Ward/Twinned Spell works when you cast a qualifying spell. It even works when an item says you cast one. #DnD
Rules text does not prevent this use of the Ioun Stone
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)
Best Answer
This functions more or less the Ring of Spell Storing.
Anyone can use the spells out of a Ring of Spell Storing or a Reserve Ioun Stone and the spells that are cast out of them are cast with all the "settings" of the original cast.
Example: Your Barbarian has a Reserve Ioun Stone and your party's Cleric casts Cure Wounds at 3rd level (as that is the limit of the stone's capacity) into it. Your Barbarian can at any time use that spell as though he had cast it barring any other restrictions such as incapacitated or raging (as you are in fact "casting" a spell), but the Cleric's DC, spell attack modifier and all other such relevant parameters are of those of your party's Cleric at the time of the cast.