A vegetarian diet for cats would be difficult to the point of impossibility.
A vegetable-based diet could be devised, but only if some chemical reagents and animal fats were added to it.
That is because cats cannot synthesize some of the essential fatty acids found in animal fat.
For example, plants have virtually none of an amino acid called taurine.
Taurine deficiency results in blindness and loss of hearing, an abnormally enlarged and weak heart, grave retardation in kittens and the birth of kittens with developmental abnormalities.