The role of the project expediter is just to facilitate communications. The project expediter is often seen as a mini-version of the project coordinator, handling only the communication part. A very common responsibility of the the project expediter is to communicate the status of the project to management.
Project expediters usually report to the project manager or the project coordinator (but they can communicate the status of the project directly to upper management).
Unlike project coordinators, project expediters are non-decision makers, and do not affect the flow of the project in any way. They are merely there just to report on the project.