Children can have a vegetarian diet, certainly, and in some cultures often do. My youngest adopted a vegetarian diet at around 5, so quite a lot older than yours, but some of the same concepts apply (though the amounts required vary).
For the most part, the best place to start is to talk to your pediatrician. They will verify that your child is growing at a normal rate, and can pay attention to various things that could crop up due to lacking certain nutrients. They also can direct you to resources for nutrition. Our pediatrician gave us some literature on what to ensure they eat, and on various options for the common things you usually get in meats.
One key thing to consider is that a few things, like B12, are pretty hard to get outside of meat; the best way to get them as a vegetarian ends up being fortified cereals or breads or similar things that have them specifically added. Vitamins can also help here.
In no particular order, some of the things you need to consider in particular:
- Iron: Many cereals are iron fortified; beans; tofu; leafy green vegetables. Peanut butter also has some.
- Calcium: Milk and other dairy products, leafy green vegetables, beans
- Protein: Milk and other dairy products, beans, nuts, tofu
- Vit. B12: Milk and other dairy products, many cereals and breads are fortified with it.
- Vit. D: Milk is often fortified with it, as is orange juice, some cereals , etc. Vit. D is also producible by your skin when you are outside, if you're outside for a large part of the day.
For protein, make sure to get it form a variety of sources; most sources do not have a full spectrum of amino acids. (Milk does have all nine amino acids, however.)
Also ensure she's eating enough; calories can be a challenge at that age.
In summary, talk to your pediatrician and see if additional testing is needed to make sure your child is getting everything they need, but it's very possible to make sure they get everything they need without meat!
Sources for some of this information: