@OP, I assume:
1. Calls are coming over normal analog lines, VOIP even,
2. Headset is plugged into pc so that calls are then recorded thru the browser, and possibly saved on a server.
I hate vendor lock in, and the OP's case is the prime reason why. If you're going down the route of finding another developer, find out if they will be willing to use
WebRTC. It's pretty new tech, but it is a draft that will be standardised ultimately. It accesses all media type hardware, requiring permission of course, whereby you can record audio/video, i.e. your calls in this case (played around a bit and got Manycam to run a looping gif; could transmit to another pc no issues). Since it runs in the latest
worthwhile browsers plugin free, you're not locked to a specific vendor who can treat you like dirt.
If your new development team is familiar with websockets then webrtc shouldn't be too much of a hassle. Furthermore, since you're doing everything on an internal network you don't need ICE/STUN servers.
tl;dr just youtube WebRTC and see if it fits your application, assuming you're getting a new dev team