Your hyperlink to the site might better be compared to a reference or footnote in an article, or a listing in a directory (the phone book, etc). You are making reference to information; you are not presenting an original work, nor a copy of an original work.
According to current case law [chillingeffects.org] (if I understand correctly), a hyperlink is not illegal unless it leads to infringing or otherwise illegal material, or is being used to disseminate such. Unless the shelter in question were committing crimes, and unless your links were aiding in such, there is likely nothing illegal about your links.
If you were claiming to be endorsed by or somehow affiliated with the shelter, then, were the shelter's name a registered trademark [chillingeffects.org], you could be held to be in violation of their trademark. But this trademarking is unlikely for an animal shelter, and you don't appear to be making any such claims of affiliation or endorsement.
The content on the shelter's site is copyrighted, but courts have held [news.zdnet.co.uk] that linking to copyrighted material is legal.
I can see no grounds upon which the shelter could win a suit against you. However, you could still be "out" the costs of your defense. In other words, even after you "won", you'd still lose.
It may be better simply to remove the link, at least for the time being.
Just my opinion, of course; I could be wrong....