Jill,
PBX is an acronym for Private Branch Exchange and manages your phone lines in your business. Almost all businesses use digital phones now, so almost all of your lines from your PBX will be digital. This would mean you could not plug a regular phone that you buy at Walmart into a digital jack.
Therefore, if you want to fax from the office you will need a POTS or plain old telephone service analog line. My guess is you already have at least one line like this which is used for a fax line.
To do what you are talking about, you will need a fax server setup which means one networked computer will need a modem and fax software. Windows does make Windows Fax, but as Microsoft is so good at doing, a lot of the things they make are not as good as 3rd party programs such as Winfax by Symantec, not that I would recommend anything from Symantec.
You then have to have clients which are capable of interacting with the fax server. You will find it very difficult to simply connect to and share the modem using the proprietary fax software from Microsoft.
This topic has come up many times on this board and most who want to PC to fax and fax to PC use networked fax machines of MFPs. Some of these are low end such as Brothers (which I am not a fan of although many on here are), to midline such as Muratec and finally up to Dell, Xerox or Toshibas which can run in the $5,000 to $7,000 range. Finally, there are companies which make standalone fax servers such as Castelle, Captaris, et. al. These are rather costly but do come with their own client software, which can generally offer many advantages over other less expensive fax servers.
But, getting back to your question. Since I don't know your network setup, e.g. Client/Server or Peer-to-peer, or if you have server software or already have a modem, etc., it is hard to give really good advice. From what you are saying though, my advice would be to have one fax line, a computer with a good fax modem connected, clients which are all networked to that computer via P2P or otherwise (this is obvious) and 3rd party software.
Many on here use Snappy Fax, which is what I would recommend for a solution. It is very good software which is inexpensive and will allow you to do what you need to do. It also has a 30-day trial period with free support so you really can't go wrong. You have to purchase the server software and the client software, but it is very reasonable. Once installed, you will have a lot of flexibility over incoming fax routing as to who can have what.
www.snappyfax.comMake sure you read what modems they recommend as that can make a big difference.
HTH