That's a very interesting section of the book. Deep down, Lyddie wants to sign the petition and even knows that she should sign the petition. The petition is requesting better working conditions for the girls in the mills. It's not a ridiculous request either. The working conditions are horrible. It's loud enough to cause hearing damage, there's fibers in the air that cause lung problems, tuberculosis is a concern, and the machines are capable of causing bodily harm to the workers.
She had thought a single stagecoach struggling to hold back the horses on a downhill run was unbearably noisy. A single stagecoach! A factory was a hundred stagecoaches all inside one's skull, banging their wheels against the bone.
The petition is seeking to fix some of those issues.
Lyddie doesn't sign the petition, because she is afraid of the consequences of that action. By signing the petition, she would publicly be associating with it and those workers. She runs the risk of being fired from her current job and blacklisted from the other mills. Lyddie cannot let that happen. She desperately needs the paycheck in order to pay back the debt that her family owes.
Lyddie does eventually decide to sign the petition but only after she is injured while working. Unfortunately, she is too late. The petition had already been sent off.