I think I’d start with Sales and see what they have to say. My guess is that the SangomaCRM modules may meet your needs, and if it doesn’t they should be able to hook you up. If you want to do it “text only” you could also submit a feature request to see it would take to get this supported.
Short of that, unless you want to open-source the package, I don’t think we’re (the user community) is going to be able to help you much.
We did this before for an old FreePBX system (PIAF - forget which version number) by using some URL or other to authenticate and then initiate the call. It worked. I think it was an Asterisk thing rather than FreePBX.
I just assumed there would be something similar available currently.
Did some more Googling and it looks like what I need is ‘Asterisk Tapi driver’.