Lyddie doesn't sign the petition to improve working conditions in the factory for a number of reasons. First and foremost, she's something of an individualist. The idea of collective action is somewhat alien to her. Lyddie is one of the best workers at the factory and she works extra hours to make extra money for her family. She's concerned that if she signs the petition and its terms are granted then she won't be able to make as much money, with dire consequences for her family.

Though sympathetic to the other women and girls at the factory, Lyddie is only working there to support her family. They rely on whatever she can send back to them. So anything that might reduce the amount of money she can send is something she instinctively takes against. Fewer hours will mean less money, and less money will mean less to send back to her family. Lyddie's refusal to sign the petition may appear somewhat selfish on the surface, but there's a perfectly valid reason behind it.

One could also argue that Lyddie refuses to sign the petition was she doesn't want to get into trouble with the management. For now, she's on the management's good side, and she wants things to stay that way. She knows that if she gains the reputation of an agitator or troublemaker then she can be fired at the drop of a hat with no compensation. Then where would she and her family be?

In chapter twelve, Lyddie and her roommates discuss the petition. While she would like to sign this statement demanding ten-hour workdays instead of the current thirteen or more hours, she decides against it. Her good work has led to increased pay for Lyddie. She is making nearly $2.50 per week, significantly more than many of her fellow mill workers. Lyddie feels that she cannot risk losing this important income. First of all, working fewer hours would result in less pay. Lyddie feels that she is her family's last hope of paying off their debts and having them reunited on the farm. Any reduction in her wages threatens her hope of saving her family.

She knows that she is well-regarded by the management. At the beginning of the chapter, the overseer shows off Lyddie to a group of foreign dignitaries as one of the mill's best workers. Lyddie knows that having her name on the petition would ruin her good standing with the mill's management. If she signs the petition, it could result in her getting fired and blacklisted, meaning she would be unable to work in any of the Lowell mills. At the time, there were no legal protections for workers. Collective bargaining in this manner was a way to run afoul of management. A worker like Lyddie could easily be fired and replaced by more compliant workers. With her family's future on the line, Lyddie cannot take the risk.

The petition that you are referencing is one circulated by the Female Labor Reform Association that aims to reduce the factory work days from thirteen hour shifts to ten hours. Chapter Twelve of the book examines Lyddie, Amelia, and Betsy's differing opinions on whether or not to sign the petition.

Lyddie is against signing the petition because she fears that she will be paid less if she works fewer hours; she wants to work as many hours as possible in order to afford to pay off her family's debts and reunite them back on the farm.

Signing the petition would also be a formal sign of affiliation with the workers. Lyddie is scared that she may lose her job if this information is made publicly known. 

In other words, Lyddie prefers the devil she does know to the devil she doesn't; it is, at this point, safer and more comfortable for her to not sign and to stick to her current work schedule

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.  

Lyddie is a very determined girl, focused on reuniting her family on their farm. Her plan, which she pursues relentlessly, is to earn enough money to pay off the debts on the farm, then return there with Charlie, her sisters, and her mother. Lyddie views the petition as a threat to those plans. The petition being circulated by the Female Labor Reform Association seeks to force the textile manufacturers to reduce the work days at the factories from thirteen hours per day to ten. Lyddie is pleased with how much money she is making compared to what she made at the tavern, and she believes that in just a year or two of working and saving, she "could go home—home free." To her, the petition threatens to "meddle with the system and bring it all clanging down to ruin." Lyddie has become very competent at running the machines, even when they speed up, and she has not had any health problems yet, unlike many of the girls who have worked there longer than she has. Because she has a specific goal in mind that she thinks the long work days will help her achieve, and because she herself has suffered no ill effects from the poor working conditions yet, she refuses to sign the petition.