Hi Gabe,
All the plug-ins are located at the same server, so Webhook should be using the same IP address as Bugzilla, so everything should work well and without any delay.
Here are a couple of things to check:
1. Are there any errors in the Webhook profile log? The log is situated below 'Save and Exit' button
2. Do you use a valid URL as 'Web Hook URL'? Can you do a POST to this URL from any other tool?
3. Will Webhook be triggered if you try any other URL (e.g. something public)?
Best wishes,
Julia.