You can buy all pets in their homes after you purchase the home. Some pets like to stay at home and cannot be walked. You can see a list of pets that prefer to stay at home under the FAQ "Which pets can't be walked?"
Walkable pets can also be purchased in the Pets section of the wardrobe. You don’t need to own a home to buy a pet in the wardrobe. You can see a list of pets you can walk under the FAQ "Which pets can I walk?"