Use AirPods with the Vonage Phone service on my Mac
If the headphones work with FaceTime and other audio apps then you can conclude the program you’re using for Vonage need some set up or you need to find a different program that respects the capabilities of macos to switch between multiple sound in and out options. 

You have done all you can by using the sound preference pane to select a  choice and it’s up to the third-party software to respect your choices.