I love backdrops from dennymfg.com, but they can be pricey and I wouldnt put the animal on it so the flooring would have to be good for the animal to sit on.
for a shelter, paper is probably easiest. I think its like $40 for a large roll, a bit cheaper for stuff that is not so wide which will probably be good enough for animals.
another option is to get some of those large 4x8 white boards at lowes *$42 here* and put that over the paper so you only have to clean the board instead of tearing up the paper as it gets dirty. its slightly reflective to which adds a cool effect to the images.
If you have the money, you could get a large sheet of plexiglass to put over that so it will be that much more reflective! But a sheet big enough is going to not be that cheap.
In the past I just had a white sheet for a background and a seperate white sheet on the ground and in photoshop I made it seamless and cleaned up any dirt spots on the floor sheet. but the wrinkles are hard to prevent.
I think usually pet portrait photogs have a good background setup and a prop to set the animal on or a nice wood flooring that compliments the backdrop. I seen one at the animal store with a ton of props, like a mailbox, a classy chair, etc.