If this is an "Expensive PAID Proprietary Service", then it kind of negates the purpose of a system like Vicidial which is Free and Open Source and runs on Asterisk which is also Free and Open Source.
That being said: Vicidial and Asterisk are fully capable of being integrated with anything anywhere for any reason. Executing an event before, during, after a call is certainly doable.
But remember the major difference here: In the Expensive Paid Proprietary world, these integrations are pre-built and licensed and you'll be paying to use them per server/user/call (perhaps all of those). But in the Vicidial/Asterisk world, you have to pay someone to Build these integrations instead. Generally costs less up-front for the customization and NOTHING down the line. Except you're stuck paying for NICE forever unless you identify it's capabilities and build THAT instead so you can skip that fee forever.
We do that sort of thing regularly. Just be sure to avoid that "wait until you NEED it NOW" thing. Tends to quadruple the price (or more). And beware those who offer to accomplish the task, but are in a location where you can not sue them. LOL
If you really want to do it right, you'll pay The Vicidial Group to build this interaction into Vicidial's core. They've done this before for things like QueueMetrics and vTiger and Sangoma AMD. Call 'em up and ask for a price. (Or us, of course, which would be my preference)