What is symbolic to Bud?
The story Bud, Not Buddy has quite a few pieces of symbolism in it; however, the question specifically asks which item is symbolic to Bud. If I had to pick one, I think it would be doors. This symbol develops in Bud's mind over the course of the story. He doesn't understand the symbolic significance at first. In chapter 5, readers are introduced to the idea that doors are symbolic to Bud. We get a little flashback story about Bud's mother telling him about doors:
no matter how bad things look to you, no matter how dark the night, when one door closes, don't worry, because another door opens.
When his mother first tells him this information about doors, Bud thinks that she is talking about a literal door; however, he begins to understand that "doors" are opportunities.
But now that I'm almost grown I see Momma wasn't talking about doors opening to let ghosts into your bedroom, she meant doors like the door at the Home closing leading to the door at the Amoses' opening and the door in the shed opening leading to me sleeping under a tree getting ready to open the next door.
Chapter 7 has another good example of Bud telling readers about his doorways of opportunity opening and closing:
That library door closing after I walked out was the exact kind of door Momma had told me about. I knew that since it had closed the next one was about to open.
His chance to find Miss Hill has closed, but the next morning Bugs shows up, and Bud steps through that doorway for his next adventure and opportunity with Bugs.