If the site does not have moderators, SE will close it, all of the posts and the site https://pets.stackexchange.com/ will be gone.
To continue there are two options, neither or both can occur, they are NOT mutually exclusive.
Migrate to codidact. Many SE communities have migrated to codidact. Some people are exclusively participating in the codidact version of the community, while others continue to participate in both codidact and SE. I personally, will gladly support and assist anyone who wants to lead a drive to migrate/copy content from Pets.SE to Pets.codidact.com. I do not have the energy to lead the drive.
Elect new Moderators, personally I find it impossible to speak positively about pursuing this option. If any members of the community want to do this you are welcome to create a new meta post. I strongly recommend that anyone involved, be familiar with Meta Stack Exchange and understand the history over the last couple of years. Additionally, understand that being a moderator in a pets community is emotionally difficult. We love our pets or we would not be here, moderating some posts can be very difficult.