I think most independent professional photographers would do it. (I got my senior portraits taken with my cat.) I know some chain stores will do it, too. In our area Picture People, Flash photography, and Celebrity Kids will all allow photos with pets.
Some of my favorite pictures are the candid shots though. If you've got a decent camera and capture a good photo yourself, you could always have that made into a card. You can do it on many online photo sites and there are lots of card makers on Etsy and eBay as well.