How can I use AMI Events to take action when a call goes to voicemail?
If you just need an event for when a call goes to voicemail you can use UserEvent[1] in the dialplan to produce an AMI event[2] with the data you need. It is up to you to have an outside application (that you write) listening for said event and taking whatever action.
[1] Asterisk 20 Application_UserEvent - Asterisk Project - Asterisk Project Wiki
[2] https://wiki.asterisk.org/wiki/display/AST/Asterisk+20+ManagerEvent_UserEvent