There is a dwemer spider in the base game - you get that from the public dungeon in Bangkorai. Then there is the other dwemer thing from Nchuleftingth public dungeon in Vvardenfell and the kitten from Karnwasten public dungeon in Summerset.

The Bangkorai pet is a quest reward. For the other two you have to find 7 different fragments each which can be combined to a runebox to get the pet. The fragments have a chance to drop from all mobs in those dungeons. Plus, they can be traded and sold (same for the runebox), so you could also just check guild traders, if you don't like grinding.

Also, just as a hint, in your collections menu, when you look at the tooltip to those pets, it tells you where and how to get them