This question is NOT:
what does John Bunyan believe about predestination
is predestination biblically accurate?
This question is:
Does "Pilgrim's Progress" take a position on predestination?
And if so, which chapter / section?
I can't point to a particular chapter but the entire book is written as if the author rejects predestination in the Calvinist sense. (Meaning irresistible grace)
The entirety of the book is written as if our choices matter - as if we have a choice. Based on that, I'd say that the book does take a position opposed to predestination in that sense, but I can't give one chapter/section that specifically makes an out and out statement.
Here is an argument in favor of Pilgrim's Progress supporting predestination.
The names of the characters defines their behavior.
Faith, Christian, and Hopeful made it to the heavenly kingdom.
Timid, Athiest, Ignorance: not so much.
Furthermore, characters do not change their name. [For example, there is, afaik, not a section which details how "Christian" used to be named "Unbeliever" and then later changes his name.]
John Bunyan's Piligrim progress in fact well supports Pre-destination and pre-choice. When the preacher gave Gospel, the Christian determined to set out to the celestial city. During the course of his journey many others also follow him, but they were not determined and lose the way. Although Christian faltered many times, God's grace was sufficient for him to reach there. The door was open for everyone (Rom.10:11-12) some have backtracked with their own choice, but those determined to put their trust in Him only would possess (Rom.10:13). Once they entered, the door is shut behind and their appears: "You are predestined to be conformed..." (Rom.8:29).