What are Betsy and Amelia arguing about? Does Lyddie agree with either of them?

The argument that your question is asking about occurs in chapter 12 of the book.  There are some minor arguments that occur during this chapter, but the main argument among the three girls deals with the factory and the present working conditions.  Betsy suggests a couple of options.  One option is for the girls to stage a walk out to force better working conditions.  The other option that she is thinking about is signing the petition.  Betsy and Amelia argue over the pros and cons of those ideas, and Lyddie is opposed to both.  At this point in the story, Lyddie is working multiple looms and making a fair amount of money.  A walk out means that she is out of work and a paycheck.  If she signs the petition, she might be black listed from all of the mills.  Again, that would mean no money.