Carrots and peas aren’t a natural part of your cat’s diet, no. You should not expect your cat to eat every kind of veggie or fruit you offer. If you would LIKE for your cat to eat more vegetation, you could try offering other kinds of fruits or veggies – or both! I have read that cucumber is a popular taste cats have. Melons except for watermelon have a similar taste and may be preferable (I strongly suggest NOT feeding any pet watermelon, as it is a diarrhetic.) Small amounts of cooked potato are okay, but your cat may prefer it to have a meaty flavor – as if it was roasted with meat or with a meat flavored sauce. Onion bulbs (the part we normally eat) are shown to produce anemia (low red blood counts), so you’ll want to stay away from experimenting with that.
If your cat is as picky as you say (and you’re still having trouble with feeding, as I see your post was 3 years ago), but also for other readers who find this post, you might try cooking your own meats for him/her/them. Many of the inexpensive to moderately expensive canned cat foods are not fully meat. The meaty chunks are often supplemented with vegetable proteins and fibers that aren’t necessarily tasty or nutritious for your cat. The only way to ensure that you know exactly what goes in is to make it yourself. An hour a week to prepare your kitty’s food isn’t that much when you consider the possible health benefits.
Not all cats puke when they eat grass. Most cats that I have known do eat small amounts of greens, even wild onion greens sometimes and few have had vomit issues associated with eating them. There is a close association between the quantity of grassy greens consumed and whether the animal is likely to vomit. So if you have a cat who likes grass and you offer it, you could put it somewhere they don’t have access to after about a minute of munching to reduce or completely prevent vomiting!
@gailcalled: Cats are completely trainable. It just takes a different mindset, just like you wouldn’t try to teach a dolphin the same way you train a dog.