I understand that you want that your Asterisk PBX use a dial-up modem as a PSTN analog card in order to make and receive calls, your Asterisk PBX convert it to SIP and put the call on a softphone.
The big problem is that Asterisk is unable to "talk" to the modem card because there are hundreds of different models and Asterisk doesn't have the drivers for each one.
There is a cheap card, the X100P card that you can use for that. Actually is a V.92 softmodem based upon the Intel 537 or MD3200 chipset.
This card is full-duplex which is required for VoIP applications and Asterisk has specific drivers Zaptel for this card.
It is possible to use a card with the same chipset however to get that the Zaptel driver recognize it you need to make a modifications to the driver and it means to fiddle with the source code of the driver.
There are also FXO ATA's that let you get the same result adding the ATA to your network.