I saw that someone had been directed here while searching for an answer to where you can find pets in Pirate 101 so thought I'd write a quick post with an answer:
Aside from buying pets in the crown shop (they cost around 2-3k crowns, I think) you can get one from the GameStop play time cards, the WalMart playtime cards, a few bundles that KingsIsle has promoted, but if you want to get them in game, the earliest two bosses I have heard you can farm for pet drops are Buster Crab in Skull Island (Ancient Tunnel, I think) and Fin Dorsal in Gullet (this is the 2nd time you encounter him). They have a small chance to drop the Orchid Chameleon.
I'd just buy one in the shop myself (and that is what I did) because I don't like farming bosses for drops (until end game, anyway), but if that's your thing, those two seem to be the first opportunity, that would be around lvl 7-9, if I remember correctly. Happy hunting!
KingsIsle developer Ratbeard just stated on the official forums that most bosses have a small chance of dropping a pet. Developer confirms! Enjoy!
KingsIsle developer Ratbeard added the following to his previous statement:
In the early Skull Island chapters, try farming Marcus, Fin Dorsal, and most of the Rat Pack bosses. (The Rat Pack side quest chain offers a ton of great boss farming opportunities!)
In Cool Ranch, try Johnny Ringo, Monterrey Jack, El Guapo, or Santa Rana.