Let’s look at cats first of all. The main reason why cats cannot eat a vegetarian diet, is because an essential part of the healthy cat’s diet is taurine, a nutrient which is found in sufficient quantities to fulfil their needs only in meat. A couple of pet food companies claim to provide a complete cat food which is both vegetarian and contains appropriate amounts of taurine and other essential nutrients- however none of these products are endorsed by veterinary professionals or pet nutritionists, and contain significant quantities of synthetic compounds to make up for the lack of naturally occurring taurine.
Also, cats are natural hunters- much more so than dogs. Dogs are hunter-scavengers, while cats in the wild subsist almost exclusively on fresh kill. If you remove meat from your cat’s diet, their hunting instinct will kick in and they will begin to search for live prey such as rats, mice and birds- which rather defeats the object of the exercise.
Dogs, on the other hand, do not require taurine to be present in any significant quantities in their diet in order to stay healthy. In terms of fulfilling your dog’s complete nutritional requirements, feeding a well thought out and researched vegetarian diet or pre-packaged complete vegetarian dog food would take care of their nutritional requirements sufficiently. Also, as dogs are hunter- scavengers, their instinct to hunt for food is much lower than that of cats, and so you are unlikely to come across any problem hunting behaviour in a dog which has not shown much inclination to hunt before.