Error: 405 (Method Not Allowed) when using POST anonymously

[Graham Bowman]

I'm trying to set up a simple Apps Script web app to receive webhook post requests from a server. The requests will be anonymous so I have deployed the web app to execute as me and anyone has access. 

This works fine with GET requests but when I send a POST request I'm seeing:

HTTP/1.1 405 Method Not Allowed

Also in the headers:

Allow: HEAD, GET
X-Content-Type-Options: nosniff
X-XSS-Protection: 1; mode=block

I can't find any documentation that indicates that scripts cannot be deployed with anyone having access using the POST method. Can anyone confirm whether this is the case or if there is a workaround?