No, cats can't become vegetarians. They are "obligate carnivores" which means they must eat meat to survive. Their bodies can't produce certain nutrients and they must get these substances from another animal that already performed the conversion process.
For instance, a human or a dog can break down something called carotenoids from plants, and our bodies convert the substance into active Vitamin A. Cats can't do that - they have to get their active Vitamin A already converted by some other animal. Arginine is another example. It's a building block for proteins that cats can't produce, and they have to get it with every meal.
Many cats love eating dog food and it can be difficult to prevent them from doing so in a multi-pet household, but keep in mind that just a little dog food can fill a cat up to the point where it doesn't eat enough cat food and will suffer a deficiency.