There are no pets of any kind in the community. Everything is tightly controlled, and everything exists for a reason and a utilitarian purpose. Since pets mostly exist to make us feel better, and people in Jonas’s community have no feelings, there is no need for pets.
Jonas comments that all young children are given “comfort objects” that are stuffed animals.
Many of the comfort objects, like Lily's, were soft, stuffed, imaginary creatures. Jonas's had been called a bear. (pp. 18-19)
These are “imaginary” creatures, so they do not exist in Jonas’s world. Elephants and bears are not the only animals that the people don’t believe exist. Surely cats and dogs would be there too.
The word exists in their vocabulary, because Jonas suggests that the children Lily did not like acted like animals. Neither of them know what the word means, but it is used to "describe someone uneducated or clumsy" (p. 5). It describes someone who does not fit in.
This does not mean there are no animals in the community. They probably get meat from somewhere. They do have fish. Jonas mentions a Fish Hatchery attendant assignment at the Ceremony of Twelve, who is happy to receive her assignment in order to help in “the important process of providing nourishment for the community” (p. 52). They are food only.
In Jonas's memories, The Giver shows him that people and animals used to co-exist and help each other.
Now he understood about animals; and in the moment that the horse turned from the stream and nudged Jonas's shoulder affectionately with its head, he perceived the bonds between animal and human. (p. 122)
Nothing exists in Jonas’s community without a purpose. They do not have pets. They do not even have real families. A family unit is creating by pairing two adults with two children for the purpose of child-rearing. There is no affection, and no love.
Lowry, Lois (1993-04-26). The Giver (Newbery Medal Book). Houghton Mifflin Harcourt. Kindle Edition.
No. When Jonas left for the community, that was the first time he truly saw wild animals. But if you are talking about pets, the Giver transmitted a memory to jonas about the relationship between the pets and people. So from that, we can tell that people in Jonas's society do not have pets.