This is equivalent to my earlier question A question about something like "shelling" in a PL manifold, but maybe more comprehensible and to the point.
Given a triangulation of the PL sphere $S^n$, is there always a subdivision (a.k.a. refinement, a.k.a. finer triangulation) that makes it shellable?
Put this way, I'm guessing that the answer is well-known.
EDIT: I quickly got two different answers, each of which seems to give just what I need. I'm more or less arbitrarily accepting Allan's.