If you lay out conflicts into general broad categories, you can organize them as follows:
- Man vs. man
- Man vs. nature
- Man vs. society
- Man vs. self
You might be able to add a few others, but I would argue that it probably fits within the above four.
Applying those conflicts to Taming the Star Runner leads me to argue that story's main conflict is man vs. self. The other ones are there. Travis struggles with his relationship with his uncle and the people of the town. Travis struggles with the country life/society. The climax of the story is man vs. nature, because of the huge lightning storm.
However, most of Travis's issues with the town, the people in the town, and his uncle are not caused by them. It's Travis's attitude about himself and about his situation that is the central focus of Hinton's novel. Travis is an emotional train wreck of a character, and much of the story is about how he handles (or doesn't handle) those emotions and emotional outbursts.
Casey is a huge help to Travis. She helps focus his energies, and through her friendship and guidance, Travis becomes a much calmer and wiser character. Taming the Star Runner is focused on Travis and his maturation, which is why I think the central conflict is with Travis and his maturation.