You should be able to install a CTA or CTU adapter on each handset that requires the dialling out feature and then install the TAPI driver for the adapter. Then assuming the database has some sort of dialler in it, you could change the device it dial's out on to be the TAPI adapter.
I've done it a for a few customers and it works quite well.
The hinge pin for the setup is really the software though, if it doesn't have a dialler, you could try to make it use the built in dialler that comes with windows, and have that set to use the TAPI driver but I can't say I've ever had to do that. Hope the database has a built in dialler. Most do from what I've found.