What events cause Waverly to run away from her mother at the market?
The most direct reason of why Waverly ran away was because she was embarrassed and annoyed at her mother. Waverly did not like her mother boasting about her and her chess prowess. She wanted to be her own person. We can see this as Waverly says these words to her mother in exasperation:
"Why do you have to use me to show off? If you want to show off, then why don't you learn to play chess?"
My mother had a habit of standing over me while I plotted out my games. I think she thought of herself as my protective ally. Her lips would be sealed tight, and after each move I made, a soft "Hmmmmph" would escape from her nose.
I knew it was a mistake to say anything more, but I heard my voice speaking...
In light of this, Waverly's embarrassment of exploding might have caused her to run away as well.