Say that I want to write a small story puzzle for Puzzling. Small sections are given per question and propose a riddle or puzzle in such a way that is not explicit (enigmatic). Depending on the solution it will determine the course for the next question in the series.
Is this permissible?
It's not 100% clear to me what you're suggesting when you say that "the solution ... will determine the course for the next question". Are you implying there's multiple possible solutions, such that the overarching story becomes a pseudo-choose-your-own-adventure? If so, then I would say no, this does not fit the guidelines for a puzzle on PSE, since puzzles posted here should have a single unambiguously correct solution. If I'm misinterpreting, read on...
The general policy for creating a puzzle is that it is self-contained and has a single, obviously correct solution. Taking that into account, you can still create a series of puzzles with an overarching story (and people certainly have in the past), so long as each "part" can still be solved in isolation. You can then add a link in each puzzle that points to the previous part of the story for users who want to catch up. If any given puzzle is dependent on prior parts, you should include the relevant information directly in that puzzle (possibly in spoiler blocks if absolutely necessary, but I would think a more general spoiler warning at the top of the puzzle with the link to the prior step(s) should be sufficient).
