You can enable sponsorships for animals, when editing or adding an animal.
Add or Edit an animal.
Scroll down to the Sponsorship Options.
For each animal you want to be sponsorable, select Yes for Allow Sponsorship.
You may also want to add content to the Sponsorship Details so that you can include why the animal needs sponsorship. With the Website Service, this will be shown on the animals individual sponsorship page.
Website Service: You will need to enable the Sponsorships feature. Here is more information on how to do this Sponsorship Settings and Options
Overview
Content Tools