I'm glad you posted this thread...I'm looking to volunteer at the Humane Society to take pictures for them. Better Pictures = Higher Adoption Rate. My biggest concern with it is the lighting. I don't have any studio equipment at all and I don't want to mess it all up for them either.
This can be a very rewarding adventure and can certainly tug on the heart strings quite a bit. I used to volunteer at my local animal shelter. Most of the photos of the animals the shelter had on their website were piss poor to say the least. I didn't just fire away from the other side of the gate. I would get inside the run and spend some quality time with the (mostly large) dogs because most of the other volunteer photogs were leary of them. I can think of only one dog that was ever slightly aggressive towards me in the beginning. After staying in the run for a while with her, she finally let me give her a pat. I usually spent 15-20 minutes with each animal so my average per night was lower than most. However, I think I was able to catch some of their personalities and the foot traffic did increase.
I wouldn't worry too much about not having studio equipment. I usually just used my speedlight and reverse fired it, bouncing off a wall. Being in such tight quarters, flashing directly was not attractive. Focus should always be on their eyes.