This guide will provide you with the locations to find the artifacts and then the Yeti Pet. In order to get the Yeti Pet in Time Adventures, you will need to find 6 artifacts in the Sub Zero Adventure.
The Snowman Head Artifact is found here. 
You can find the Sword Artifact on top of the waterfall. 
The Egg Artifact is found inside the waterfall. 
You can find the Penguin near the other side of the lake past the waterfall. 
The Viking Helm Artifact is found at the Yet’s base. 
Now getting the Frozen Claw is somewhat difficult. First, you need to put the core in the teleporter and when the teleporter becomes unstable, you’ll be thrown into the avalanche. 
You’ll find a door there and you need to head in the direction opposite the door. If you get teleported, you have to quickly head out of the door and find the claw. Once you find the claw, you just need to touch it to get it. 
Once you’ve gotten all 6 Artifacts, head back to the lobby and click on the pet to get it. 
Here’s what the pet looks like in the game. 