I did a check of animal companion classes recently (to build a party of all halflings riding leopards), and what I found is:
Ranger, Hunter, Druid, Cleric(Animal), [Mad Dog]Barbarian, [Beast Rider]Cavalier, [Sylvan]Sorcerer, [Sacred Huntsmaster]Inquisitor and [Wildland]Shaman have access to all animal companions. I think Shaman(Nature)also has full access at 16.
Paladin, other Cavaliers, [Arcane Rider]Magus, [Sohei]Monk, [Bloodrider]Bloodrager and Oracle(Nature) have very limited options. Mostly horses, Bloodriders only get Smilodons, Nature Oracles only get horses or wolves.
Alchemists, Arcanists, Bards, Fighters, Kineticists, Rogues, Skalds, Slayers, Warpriests, Witches and Wizards have no animal companion access.
That's 11 classes with no access, 5 with limited access and 9 with full access to animal companions.
There's a decent chance I made a mistake somewhere, though, I wasn't particularly diligent with this.