If you can coax out a CSV file from the phone system say on a daily or several times a day basis, then QuickBase Sync for CSV can poll Dropbox and a couple other similar services each hour to see if there is anything new to automatically import. I think that is is more of a question for you phone supplier to see if it can create that CSV file and if it can be automated into DropBox.
Yes it's possible, but the leads are time sensitive so immediate input is critical. I think you are right about it being a question for the Dialer provider. Just curious if anyone out here in the forum has experience doing this. I did find some info that the dialer has an option for web forms. Maybe this is how it can be done?