Vicidial is already PHP. Adding a framework to Vicidial Core would slow down Vicidial, which is not an acceptable situation for most. That being said: There is some vagueness about your request that implies a non-technical background:
"Integration with PHP Symfony" ... symfony is a framework for php. It would not be so much an "integration with" as a "rewrite in" symfony. Unless you are trying to discuss integrating Vicidial with an application other than Vicidial, and that application happens to be written in symfony.
That's doable. But not related to Symfony or the framework of the application at all. Vicidial has integration methods built in. Vicidial can send and receive data in many different ways. For integration with ANY application, you merely need a data map, a trigger, and a filter.
What data do you want to send to and/or receive from Vicidial? Where is it to be stored in Vicidial and the other application? What will cause this data exchange to occur (button press, call start or end ...)? Are there certain situations where it happens and others where it should not (such as the end of a call initiates the transfer but ONLY if there was a SALE disposition).
All that being said: This is not so much a general discussion as it is a support request. I'll move it to the support board.
Welcome to the party!