Hi J.,
I'm a mom to a 6yr old ovo-veggy and the class I teach at our day care is the 1-3yr olds. Like it was said before, at 3 it's not so much that she'll feel like her food is different. Most kids at that age aren't heavy omnivores. All 6 of our kids are required to bring their own food for the day. I see lots of yogurts, PB & J, fruit of all sorts, cheese, crackers, edamame (mine prefers it in the pod so she can have the fun of popping them out), soups, noodles, beans & rice, and left overs from dinner the night before.
Please make sure to let her teacher know of your dietary requests. Many kids have food issues of some sort these days. I always keep some cupcakes in the freezer for those party emergencies. Occasionally, a child in my class may run out of food in their lunchbox. I keep a stash of Cheerios & banana chips in my desk for that reason. You may ask the school if you could leave something like that especially for your daughter in case she runs out, or her lunch spills onto the floor.... happens a lot!
Don't feel shy about your daughter's needs. Teachers are used to working around nut, dairy, and other food allergies. If you make sure you're prepared for those "treat days" your daughter won't feel any different.
Enjoy!