In Lyddie, what is an example of symbolism in Chapters 1-10?

An obvious symbol throughout the book Lyddie is the bear that Lyddie stares down in chapter 1 and which she remembers often. Lyddie's encounter with the bear represents her ability to deal successfully with challenges and danger.

Another interesting symbol occurs in chapter 4. When Lyddie starts working at the tavern, she gets to know Tryphena, the cook. As Lyddie churns butter one day, Tryphena tells her the story of two frogs who fell into the pail of milk. One drowned immediately, but the other one kept kicking and kicking until it formed a pat of butter on which it could float. Tryphena explains the symbolism of the story: "Some folks are natural born kickers. They can always find a way to turn disaster into butter." The first frog could represent Lyddie's mother, who gives up on life after her husband leaves the family. The second frog represents Lyddie, who doesn't give up and works with all her might to save her family's farm and to try to make a better life for herself and her siblings. 

The frog story brings to Lyddie's mind the private joke between Lyddie and Charlie: "We can still hop." Their mother had written to them with her incorrect spelling, meaning to say, "We can still hope," but instead writing, "We can still hop." Lyddie's mother's approach, to hope vainly without "kicking," results in her drowning in her grief. But Lyddie, like the frog, can still hop. She kicks and takes action, and that keeps her afloat until her life improves. 

When Charlie comes to visit Lyddie, she finds little to talk about. After he leaves, she wishes she had told him about the frogs. Connecting Charlie with the frog story symbolizes that he, too, will survive because, like Lyddie, he can still hop.

The way that I interpret your question is that it is asking for a single symbolic item that is found in any of the chapters 1-10.  

I would pick the bear encounter from chapter one as a good piece of symbolism for Lyddie.  The bear is symbolic of the struggle and hardship that Lyddie will have throughout the novel.  The encounter is also symbolic because it foreshadows Lyddie's courage, determination, and fortitude in the face of incredibly difficult circumstances.  Lyddie is a fighter.  She is not somebody who is going to back down from a challenge.  She will fight for what is important to her despite seemingly insurmountable odds.  The bear is symbolic of those insurmountable odds.  At no point should a thirteen year old girl be able to fight and scare off a bear.  Yet that is what she does, and symbolically, that is what Lyddie does for the rest of the novel. 

What are some examples of symbolism in Chapters 12-14 of Lyddie?

One symbolic image is money.  Through this point in the novel, Lyddie has been extremely focused on money.  If anything, her obsession with money is increasing during these chapters.  Lyddie has become a great worker, and she refuses to complain about the speed up or working conditions.  She refuses to sign the petition, because she doesn't want anything interfering with her ability to earn money.  In chapter 12, Lyddie receives a request from her mother to send more money.  In chapter 14, Lyddie reluctantly encloses a single dollar in her return letter to her mother.  The act is symbolic of Lyddie's negative character development.  Early in the novel she gave all of her money to Ezekial, a man she barely knew, because it was the right thing to do.  Now, Lyddie can barely spare a dollar for her own mother.  

The other symbolic image in these chapters is the image of slavery.  In chapter 6, Lyddie saw parallels between her life at the tavern and Ezekial's life as a slave.  Lyddie feels that she has gotten away from being a slave because she is being paid so well in Lowell, but the girls that Lyddie works with feel differently.  Betsy is the most vocal about their treatment, and she even sings a song lyric that is about factory girls being slaves.  Lyddie will eventually see that Betsy is right about how the factory owners see and treat the factory girls.