Can only get cat via "cat machine" in your tent!!

Edit: and from quests and buying dlc.

Deer and rabbit will come to the shop later in the game after you get letters from your boss via post bear!! so you will need to wait until day 10/15 ish.

You can buy critters from Lee. He has deer, rabbits and birds. He's a Spirit Bear that appears later on in the game. Sometime deer/rabbits may arrive by post from the Scoutmaster. But that rarely happens. Your best bet is to keep doing quests to unlock the Spirit Bear, Lee Berry Dennings.