I am currently working on a series of mini-puzzles I’ll be posting on the main site, and I have an idea for a mechanism that utilizes features of users all across the Stack Exchange network, specifically their avatars and user IDs. Is this allowable? This meta post suggests that it is fine to use specific usernames for story purposes, but doesn’t provide any insight on using the users themselves.
I would highly recommend that you don't use specific users as part of a puzzle, for two main reasons:
Preservation: One of the many goals of this site is to be an archive of high-quality puzzles that can be solved at any point in the future. That's why we have the general guideline that puzzles not rely on external resources. Users can change their display names or avatars, or even delete their entire account! (And this isn't just a hypothetical - I can think of an instance of each of those from a prominent user off the top of my head.)
Obscurity: Having puzzles that rely on knowing particular Puzzling users may be problematic even if that information is still unchanged. The people who you refer to may not be anywhere near as prominent in the future. And additionally, people who aren't members of Puzzling will have very little chance of being able to solve the puzzle. You'd be creating something that is only solvable by people in this particular clique of active members right now. I think we should try to encourage participation from new people, and this seems to me to do exactly the opposite.
So even if you technically could do so, I think it would not be a good idea, both for the quality of your puzzle and for the goals of this site.