Why does Lyddie not sign the petition in Lyddie?
The petition that your question references is a petition to improve working conditions in the factories. While the girls are paid to work in the factories, they are worked extremely hard in conditions that are not always safe for the girls. These difficult working conditions often result in injuries to the factory girls.
While Lyddie might agree with the petition in concept, she doesn't sign the petition for two reasons. Fear and selfishness.
Her fear is a legitimate fear. Girls that have signed the petition are blacklisted from the factories. That means that they are fired, and other factory owners will refuse to hire them. Signing the petition basically guarantees job loss and loss of income. Throughout the novel, Lyddie is motivated to be financially independent and pay off her family's debt. She simply cannot bear the thought of being fired.
Selfishly, Lyddie doesn't sign the petition, because as more and more girls get blacklisted, Lyddie gets more and more work. That increased work load translates to increased income for Lyddie.