To get the Pet Sitter Achievement, you need four pets. Pets are unlocked via mementos which can either be found during the levels or bought in the lost and found.
Pets[]
Speak to Rucks after you get the Ragged Hood from The Workmen Ward .
Speak to Zulf after you get the Plush Pyth from Pyth Orchard .
Speak to Zia after you get the Anklegator Egg from Roathus Lagoon .
Speak to Zia after you get the Pecker Nest from Colford Cauldron .