Freepbx ticket system (call center)

I would recommend APIs are how you would connect them.

FreePBX Generic API instructions I linked to you above. I googled vtiger APIs and found this:
https://help.vtiger.com/article/147111249-Rest-API-Manual

If you see the APIs you need to do the things you want in vtiger, then you can move on to programming what/when you want FreePBX to do when. What you are asking to do does require you to develop. I recommend reading and trying things out. You can always ask specific questions on how to do things here, if you get stuck.

To get you started here is some additional recommended reading material: