Vicidial will work with any trunk that will work with Asterisk. So actually, your goal is NOT vicidial related as all you need to do is figure out how to make your trunking method work with Asterisk.
If you search for a method that will make a 56K Voice Modem work with asterisk, however, you may come back disappointed. But Digium (the corporate sponsor/owner of Asterisk) makes hardware for this purpose. There are also other companies that make similar products. Similar to a modem, but often with up to 4 lines, still capable of just plugging into a PSTN wall jack (ie: the phone line your 56k Modem would plug into for faxing).
The downside of this approach is that now you have extra hardware inside your box and it must be 'configured' at boot time and you can't use that hardware with another box (without yanking it out of this box).
Grandstream makes a standalone device (called a Gateway) in 4, 8 or 12 port designs that will JUST convert a the wall jack PSTN to a SIP trunk. Then you would consider the Grandstream device a SIP carrier. Even though it may sit right next to the Vicidial machine .. it is not IN the Vicidial machine. So when you kill your HD, you can fire up the "backup server" and go right back to work. In fact, properly configured, you can pull phone lines from the wall, pass them through the internet, and hit another Vicidial machine on the other side of the world (so it does not actually NEED to sit right next to the Vicidial server at all).
When purchasing the Digium module/card, beware that you get the "Trunk" instead of the "Station" modules installed so you can connect to the wall jack. The other would be designed to allow you to connect a standard phone to it (not useful for making calls! It's for agents to connect to the Vicidial server!).
http://www.amazon.com/Digium-TDM410P-Ba ... gy_e_img_zhttp://www.metrolinedirect.com/grandstr ... 7Aod1kMAEwLinks are not a recommendation to buy that particular product from that supplier. Just an example of what you could google or eBay.