As others have suggested I agree that a bit of randomness will help you out a lot. However, I wanted to over some additional suggestions.
The randomness only really provides a solution to the "fairness" of which animals are seen. However, given your specific scenario I would argue that there should be a bias towards certain "items" getting priority. And that is, I think animals that have been with the shelter the longest (and supposedly the least desirable) should get a bit more promotion than the new additions.
To resolve this I would potentially look at allowing the user to choose different "sort orders" for the list. By default you could have the random sort order, as discussed. And also have a sort order for the length that they have been with the shelter. The longer they have been there, the further up the list they will appear.
The benefit of this is it caters for your viewers that are looking for animals that don't get much love (i.e. the ones that have been with the shelter the longest).
As a side, personally I would be tempted to try and guilt trip the viewers. I would have a sort order called "Most Unloved" (or something along those lines) that would give priority to the ones that have been there the longest (as mentioned above).
I would also make sure every item has a "X days at the shelter" info too, if you don't already have something like that. At the end of the day, your site is there to try and "sell" (convince, not profit) the animals, so you should promote those that need the help more. The cute, friendly, well behaved animals will sell themselves, so try to promote the ones that don't get much attention.