You will find the companion pet cheevos in the main section of the Character cheevos.

The Pet Cheevos go up to 250 pets. With 333 free pets, this number is easily achievable without spending credits.

A Loyal Companion – Collect your first Companion Pet.

Companion Compatible – Collect 10 Companion Pets

Critter Collector – Collect 25 Companion Pets

Zoo Keeper – Collect 50 Companion Pets

Master of the Menagerie – Collect 75 Companion Pets

Naturalist – Collect 100 Companion Pets

Animal Rescue – Collect 250 Companion Pets

Open up your Character screen and select the ‘Companions’ tab on the left side menu. This tab will only show up after you have collected your first Companion Pet.
You can drag the companion icons to your bar so you can summon your favourite pets without having to find them every time! 

• Some pets can also be looted from the special world event 4 hour minion adventures, as these adventures have not been running for over a year, I have not added this info to the data, unless there was no other known way to get that particular pet.
• Some pets can be bought from multiple vendors in the world. All of these are also in the Rift Store, except the single vendor in Uttila East, Vostigar Peaks.
• Yaki was datamined as being part of the 2020 Fae Yule event. As far as I can tell, this pet was renamed to Snowfall.
• There are a few pets that are only available from the special Fae Yule 4 hour minion missions – These pets are currently tagged as unavailable, as the 4 hour event missions havn’t been turned on since April 2022.

Companion Pet lists from other sites. Some of these were very helpful in my quest for information about the companion pets.

Rifthead – Archived.
Telarapedia
MMO2GO
Magelo
Zam Rift
Rift: Planes of Telara Wiki