I believe that Doodle can be used to illustrate both internal and external conflicts. Externally, Doodle struggles with regular every day activities and life itself. He is frail, and all throughout his life, his family is more or less expecting him to die. They have moments of joy. For example, the family is ecstatic when they learn that Brother has taught Doodle to walk, but even Doodle himself knows that he isn't quite capable of keeping up with everybody else. The simple act of moving from point A to B is a huge struggle for Doodle. That's why he is so concerned with being left alone. He says "Don't leave me" a total of four times in the story.
Physically, Doodle struggles to keep up with everybody, and I believe he struggles with that internally too. He wants to do everything that Brother does. He desperately wants to keep up with his older brother, but he knows that he is limited. When Doodle was learning to walk, it took weeks and weeks to accomplish a few steps. There had to be moments when Doodle simply wanted to give up. If that is too much of a hypothetical for an internal conflict, then Brother is a good character to use. He openly admits that he struggles with his pride. He's embarrassed that he has a brother that can't walk, so Brother decides that he will teach Doodle; however, he knows that his motivations are selfish. There's also a sense that Brother struggles with feelings of guilt over how he treated Doodle.
The internal conflict in the story comes from our narrator, Doodle's brother. He struggles with guilt over how he treated his disabled brother. He feels bad for playing mean jokes on him and for pushing him beyond his limits. The guilt especially plagues him when he recalls how unconditionally Doodle loved him no matter what he did to him. Doodle is grateful when his brother saves him from drowning, not even realizing that it was his brother who pushed him in.
The external conflict is best seen through what Doodle faces in the world. He is disabled and he must battle to be normal and he must battle to win over the heart of his older brother. Doodle pushes his body to the limit just to please his brother. He cares nothing for being "normal" only for being loved (which could also be an internal conflict). He deals with being outwardly rejected by the world, although he seems not to mind.