So far I've found:
Most Druids
Most Rangers
Most Hunters
Most Paladins
Most Inquisitors
Are there any more?
Most Inquisitors? Is that counting the Animal Domain? If so then you should add most Clerics. There is also at least one archetype for Fighter, Cavalier, and Barbarian.
Most cavaliers, of course.
A couple of barbarian archetypes
One bloodrager archetype, blood rider IIRC
Clerics with the animal domain
Construct rider alchemists get a construct mount
Wildblooded sylvan sorcerers get an AnC
Plus anyone with the animal ally feat.
Even with out feats, there is at least one option for almost all of the classes to get a pet.
Also add lunar & nature oracles with appropriate revelations, and reliquarian occultists can get the animal domain.