My daughter has three open round wounds on her leg. I think she has scratched her bug bites. What would cause the circular wounds?

Possible Infection. If these lesions were originally insect bites that she has been scratching they may have become infected. If they are painful, red, warm or she feels feverish she should see her healthcare provider.
Excoriation! Scratching will remove an area of round raised inflamed tissue, giving you round wounds. I see this always in my "scratchers." danger is secondary infection and delayed healing, which can cause hyperpigmentation and scarring!
Circular wounds. Always consider tick bites and subsequent lymes disease with circular wounds that appear with a "bullseye" type pattern. With this in mind, antibiotic agents therefore may be indicated. To prevent the onset of worse pathology with Erythromycin being a good antibiotic if a suspected tick bite prior to gross cellulitis/infection.