First a search engine for animal related non profits near you. Then get out of the computer, take your car, drive there and talk to them.
These folks are usually not very well connected to the internet. Even if they are and you can email them, don't.
Go talk to them in-person. We as computer scientists, ops engineers, designers, etc. think the world turns around the internet and that one can solve anything with it. That's wrong. Or at least it's not completely true. But it can help.
I've been looking into making an impact to non-profit for a while now and I my professional (technical) skills are surprisingly useless for 'most' of these non profits.
But there are some and I know it is possible to go talk to them and show them that technology can make their life much easier.
The non-profits you will visit most likely have more urgent needs like finding placements for their animals, buying blankets, etc. Things that technology can barely help. They more likely need advocates, people who know how to raise fundraising more than someone who can revamp their website or make sure it has SSL and is deployed automatically.
However, don't feel down. I am absolutely convinced you can make their life easier and you should. Especially since you seem really motivated by this. However, expect to work a bit with them before: cleaning cages, walking the animals, etc.
Then maybe you'll have an idea that can bring technology in their life to empower them and make them much more efficient.
Technology is a tool to help humans achieve a goal it is not an end by itself.
Well, I digressed a bit :D I hope you'll forgive me ;)