You can. Dog foods such as Bakers Complete contain vegetables, but also contain meat. However, you can get vegetarian foods. But why would anyone want to feed a natural carnivore only vegetables?
No - cats do not get all the nutrients they need from a purely vegetarian diet. Cats are not vegetarians. I think it would be unfair for anyone to impose a diet you have chosen for yourself onto an animal which has no choice in the matter - particularly when it is a completely unnatural diet for them.