I need help forming an essay on The Things They Carried. I want to write about how being a coward contributes to the actions throughout the book while relating to courage and reputation. I want to form a strong thesis based on that. Then, I need ideas on what to write about for each paragraph and some quote options for each paragraph as well. There are some good ones at the beginning of the work that show that the men were essentially "embarrassed" to go to war and thought they would lose their public image. Another one is when he goes to Canada to fight in the war—I want to get into that too.

The first step you need to take is to refine your thesis. Do you mean that the main character was afraid of being a coward, and that's why he went to war? If this is your thesis, make a strong claim using this idea.
Then, break down your argument into sub-arguments. Perhaps you can trace the main character's progression as he tries to figure out whether to fight in Vietnam or go to Canada (you might also bring in the other soldiers in the book). What are the steps he goes through? I can't provide each of these sub-arguments for you, but I can give you an idea of what they might look like.
A possible thesis might be something like: Though he is opposed to the war, the main character decides to fight in Vietnam essentially because he is a coward and is afraid of appearing weak in front of others.
Your first sub-argument might be that the main character is opposed to fighting in the war (back this up with quotes/evidence from the book and make this claim more specific by providing the reasons he opposes fighting in the war).
Then, you could argue that the second stage the main character goes through is ____ (fill in this idea and back it up with quotes/evidence from the book; you can also make a claim about the other soldiers if you choose to focus on them).
Next, you might say that the main character's decision to go to war is essentially not an act of bravery but, paradoxically, an act of cowardice. You can look in the story "On the Rainy River," in which the main character says at the end of the story, "I was a coward. I went to the war" (58). This story explains his decision to fight in Vietnam.
Be sure to have a conclusion. In this paragraph, you might wrestle with the paradox, or apparent contradiction, of going to war but feeling like a coward. What have the main character and the other soldiers sacrificed by going to war, and why are they cowards? What would have been less cowardly, in the main character's mind?
These are only some ideas for you to consider; the ideas in the essay have to come from you, but this answer will give you an idea of an outline you can adapt.