It sounds like the Plover software works at a (keyboard) device driver level, and so can be used to enter steno-to-text in any desktop application. Thus I would arrange things this way:
Put Apache on your reporting server, and set up a web application on there which shows a text box. You can use sockets (ideally) or AJAX (as a fallback) to transport your text from a browser to the server. This can then be sent out to any number of clients on a different page, probably via a database as an intermediate store. AJAX requires frequent polls and so is inefficient and slow, but on a LAN with a small number of users it would still be OK. Sockets are better but need a library to implement - take a look here at a PHP example.
Take a look at this answer to understand the different ways a browser and server can communicate (especially the section on HTML5 Websockets). Pusher is mentioned - that makes it really easy, but if you are broadcasting on a LAN it seems pointless to need the internet. I'd do it myself, for what it's worth.
If you want to stick with AJAX, jQuery, MooTools or Prototype is fine. If you want to use sockets there are several libraries that will use sockets first, and then fall back to a variety of technologies (long polling) and then finally AJAX. This will depend mostly on browser support for these various things.
I believe web sockets need a server component for which Apache is generally considered unsuitable. The first link I gave, for the Ratchet library, looks like it has its own listener component. Perhaps a good first step would be to work through the demos, so you can understand the technology and customise it for your needs?