One of my PC's wants to have a fox- just a regular albeit tamed fox- as a pet, and frankly I want to let him have it. Does Pathfinder have any specific rules concerning non-class-feature pets, or do I just use the rules for Handle Animal?
Handle Animal covers everything you need to know.
The Handle Animal skill allows the player to train animals to learn specific tricks and abilities as defined in the skill description, as well as handle them as pets. You can also teach general purpose commands if you want the animal to perform a specific role in the party.
You are trained at working with animals, and can teach them tricks, get them to follow your simple commands, or even domesticate them.
-Core Rulebook, Skill Descriptions, Handle Animal (emphasis mine)
Ultimate Equipment also covers the general list of pets a player can purchase as well as their respective prices. Statblocks are not provided, but instead must be located in the Bestiary entry of the desired creature.
Pets obtained through role play or purchase are handled through Handle Animal, and there's various equipment for the pet and the pet owner in the books.
That noted: If the player brings the pet adventuring, and there are no house rules or GM leniency in play, the pet will have a very difficult time surviving after the first couple levels. If nothing else, Environmental hazards and Area-Affecting spells will not be kind to an un-leveled animal.