Vegetables by themselves would not be nutritionally sufficient for your dog, Lindsay. Since dogs are omnivorous, they can be fed a balanced vegetarian diet, but it’s important to keep in mind that proteins from high-quality animal sources contain all of the essential amino acids that dogs require, while proteins from plant sources may contain only some of these essential nutrients.
Choosing a plant-based diet requires careful planning and supplementation. Vegetarian foods can be well-balanced with egg and milk products, while vegan diets are more challenging to balance. A vegan diet may also be deficient in amino acids such as arginine, lysine, methionine, tryptophan, taurine, and may be lacking in iron, calcium, zinc, vitamin A and some B vitamins.
If you do decide to go with a vegetarian diet, make sure you work with your veterinarian to find supplements that meet your dog's needs, and have your vet frequently monitor your dog's nutritional status to prevent any possible health problems. If you choose a commercial vegetarian diet, check the label to make sure an AAFCO (Association of American Feed Control Officials) feeding trial was done. This will indicate more about the diet’s nutritional completeness.
A better option may be to stick with a high-quality, meat-based food that you can supplement with the carrots your girl seems to love. Many people feed their dogs carrot sticks as treats, and as long as 90 to 95% of your dog's daily calories come from her regular food, nutritional imbalances should not be a problem.