If a postdoc requests a private office when negotiating an employment contract, is there a possibility the request would be granted?
The big issue in having a private office is available space.
Most departments have a limited amount of space available, and what is given to any particular research group is correspondingly restricted. There may not be a private office available to the group, in which case it wouldn't be possible to accommodate such a request. In addition, if there's a more senior person, then usually such an individual would have a "higher claim," so you might not get one.
Of course, if you never ask, you'll never know.
Sure there is a possibility. I have seen PhD students have private offices in some places. In others, even Assistant Professors had to share. And my current employer famously had a (short-lived) initiative where they wanted to implement a hot-seat scheme for everybody except top administrators (it did not go over well with faculty).
However, in practice you may not so much negotiate but ask for it. If they have the room they will be happy to oblige (since it costs them little), and if they don't have the room they will hardly be able to change this to accommodate your request. Further, note that room situations change, sometimes quite quickly - you may have a comfy single office when you start, and a few successful grant applications later you need to share the same space with two others. It is unlikely that your PI will guarantee you in writing that you will have and retain a single office at all times.
One of the comments also asks whether there will be "friction" if you have a single office and others on your level need to share. This is of course possible, and it depends a bit on your personality whether you would mind this (I personally wouldn't). However, there's usually advantages and disadvantages to all rooms. Your single office will likely be of the "small dark room in the corner" variety, while their three-person office is surely larger, and often nicer - so it's not a straight up "you got a much better deal than others". Further, if you always worry that others might be envious of whatever you manage to negotiate on top of the standard package, then why bother negotiating at all?