[RPG] Can a familiar be targeted by the Awaken spell

The Find Familiar spell is currently the only method for a wizard to obtain a familiar, and it lists the creatures available. However, a large number of creatures in the Monster Manual, such as the pseudodragon, imp, or quasit, include the 'Familiar' variant. Note that a Warlock with the Pact of the Chain feature can obtain many of these creatures as a familiar.

What you have to remember is that variant monsters, like monsters, are designed for the GM to use to make enemies more interesting. The Familiar variant is a monster variant, just like the troll's Loathsome Limbs variant or the Genie Powers variant. They're for GM use rather than player use.

The Mage NPC in Appendix B of the Monster Manual also has a Familiar variant, which says:

Any spellcaster that can cast the find familiar spell (such as an archmage or mage) is likely to have a familiar. The familiar can be one of the creatures described in the spell, (see the Player's Handbook) or some other Tiny monster, such as a crawling claw, imp, pseudodragon, or quasit.

So the Familiar variant is for GMs to create more interesting NPCs, rather than to provide players with additional options. Of course, with your GM's permission, you could obtain a more interesting familiar. This would probably involve actually finding such a creature and somehow forming a bond with it. But this relies solely on your GM to allow and arbitrate.

It's also worth considering that allowing a wizard to obtain one of these more powerful creatures somewhat invalidates the Warlock's Pact of the Chain.

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)