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Abilities could be something like
Passive: Animal Mastery
Beasts level up for every enemy champion kill they or their master makes. Each level up raises their health, AD, AS, armor, MS, MS, crit.
Q: Summon Falcon
Summons a Falcon that prioritizes enemy champions. 10/20/30/40/50 (would have fast attack speed to make up for low damage)
W: Summon Wolf
Summons a Wolf that seeks out the weakest nearby enemy champion. If no enemy champion is nearby, it follows it's master. 30/50/70/90/110
E: Summon Bear
Summons a bear that will stun nearby enemies who have high hp. Leveling this skill raises HP instead of AD and they recieve bonus HP based on a percentage of their master's max health. 100/150/200/250/300
R: Call of the Wild
Beast Tamer and all summoned beasts in play recieve health regeneration, 20% crit rate, and bonus max HP.
I know it's probably not balanced, but Riot could do the math and make it balanced if they approve this idea. Also, it probably looks like this champion wouldn't take skill to play, but I think it would take a lot of strategy to plan where and when to summon each beast. Do you send in a Falcon to chase that fleeing Lee Sin? Or do you summon a Bear to protect you from that Darius about to hack you into bits?
Also, to make him more balanced, he could summon only 1-2 beasts at a time, but he can recall a beast at any time.