I talked to my cousin, who is on her way into Joplin with food (BBQ) and other supplies (they live less than an hour away). She says that, although the people are starting getting what they need, the local animal shelters have lists of requests here. I'm sending her down a check to use as she sees fit for the effort but she says it'll go for dog food, cat food, and kitty litter. I figure it'll be easier for her to get supplies and drive them in than for me to mail them. You can donate to the Joplin Humane Society directly here.
If you don't have someone like my cousin in the area, you can mail supplies to:
: 140 Emperor Lane Joplin MO 64801
ETA: They also have a wish list on Amazon: http://www.amazon.com/gp/registry/wedding/9NXFU5BUZCLKThey are in need of: Flea and Tick products (animals are covered in ticks from downed brush); Cat Litter, and money for veterinarian care for the displaced pets.
If you are in the area, they are looking for people to foster animals who have gotten separated from their people. They have a sign up sheet here: http://www.facebook.com/topic.php?uid=144041035612022&topic=330
ETA: Donation link page: http://www.joplinhumane.org/donate.html