Assuming you already have the IP phones, you need two things. A PBX server (for the VoIP stuff), and a SIP trunk with a block of external phone numbers.
Start with the PBX server software, there's several free/open-source implementations. Once you're comfortable with it and have internal calling good to go, then you can spend on the SIP trunk and number blocks.
What do you mean by "a block of external phone numbers?" We'd like to simply have our own internal numbers ideally, nothing to connect to the regular phone network.
SIP providers usually sell numbers in contiguous series for businesses. For example, if your company buys a block of 50 numbers, the SIP provider then allocates XXX-5100 to XXX-5150.
But since you're keeping this strictly internal, you don't have to worry about that.
Step 3: unfuck the SIP settings, then email both HR and their supervisor to throw them under the bus. Also covers your ass for step 4.
Step 4: Route the manager's calls to a disconnected number. When they come knocking about their phone not working, tell them, "No, you should be able to dial out, unless someone changed the SIP trunk settings and didn't tell me."