Fabric.com has a great selection of Faux Fabric to start with.
Here's a tutorial on how to make envelope pillow covers. Just find a pillow form or piece of foam large enough to suit your dog.
Joanne Fabric stores have lots of remnants, classified by what type of material. They are always having sales.
I make my cat's ones from old bathrobes I get at thrift stores. Cut out a big enough rectangle to fold over right sides together and stitch around leaving one end open. Then for a dog I would fill with cedar chips for flea resistance. For a bigger dog, you might look for some faux fur lap blankets.