The conflict in Bud, Not Buddy is Bud's situation after his mother dies and he is left in the care of the state. He is placed in a foster home with an abusive family and runs away to find the man he believes is his father.
The rising action follows Bud's escape from his foster family and his days "on the lam". He is helped by an "adopted family" on the food line at the mission in Flint, Michigan, and a kind librarian at the local library. Bud then runs into his street-wise friend Bugs, and the two spend time at "Hooverville", a makeshift enclave where the homeless gather. It is in Hooverville that Bud encounters the unforgettable Deza Malone, and receives his first kiss. Bud then heads for Grand Rapids where he thinks his father is, getting a ride from Lefty Lewis, a mysterious man who delivers blood to the hospitals.
The climax of the story is when Bud reaches his destination, and meets the boys in the band and the man he believes is his father, Herman E. Calloway.
The remainder of the story represents the falling action. Bud is taken in by the band and especially by their lead singer, Miss Thomas. Calloway is hesitant to accept Bud, until it is revealed that he is in fact not Bud's father, but his grandfather. The story ends on a positive note, with Bud seemingly having found a home at last.
I don't agree with the other answer to some extent. While I do agree that the conflict is Bud's state, being on his own, and that the rising action is his journey to find his dad, I don't agree with the climax being when he walks into the Log Cabin. The most anticipated heart pounding moment is when Bud yells out his momma's name and HEC is bewildered and stagering into his bedroom. Everything after that is the falling action. The resoloution is him staying with the band and finding his home.