It's dangerous and probably futile to try to feed a cat a vegetarian diet. If denied meat at home cats likely will go hunting for mice and birds and whatnot. If unsuccessful, they may suffer serious health problems. Cats require specialized micronutrients abundant in meat but not readily available from other sources. Other nutrients they obtain primarily or solely from meat include arachidonic acid, vitamin A, vitamin B12, niacin, and thiamine.
Cats also require more protein and less fiber than a typical vegetarian diet provides.
You can buy supplements and veggie cat foods to make up for the deficiencies in a veggie diet to an extent. But many pet experts, including some vegetarians, say you'd be exposing your cat to a foolish risk. If you're interested in vegetarianism primarily for health rather than ethical or ecological reasons there are less problematic ways to improve your pet's diet. You could get a better kind of pet food, for one thing. Donna Dunlap, co-owner of Chicago's Park View Pet Shop, recommends the Precise, Solid Gold, and Abady brands, which among other things are preservative free.