We are using Classic Mode for the app pool. I was getting the 404 even though we had verb="*". We had to go in and specifically allow PUT requests for the app. The web.config was modified in the < handlers > section which is within the < system.webServer > section with the following:
<add name="ExtensionlessUrlHandler-ISAPI-4.0_32bit" path="*." verb="GET,HEAD,POST,DEBUG,PUT" modules="IsapiModule" scriptProcessor="%windir%\Microsoft.NET\Framework\v4.0.30319\aspnet_isapi.dll" resourceType="Unspecified" requireAccess="Script" preCondition="classicMode,runtimeVersionv4.0,bitness32" responseBufferLimit="0" />
<add name="ExtensionlessUrlHandler-ISAPI-4.0_64bit" path="*." verb="GET,HEAD,POST,DEBUG,PUT" modules="IsapiModule" scriptProcessor="%windir%\Microsoft.NET\Framework64\v4.0.30319\aspnet_isapi.dll" resourceType="Unspecified" requireAccess="Script" preCondition="classicMode,runtimeVersionv4.0,bitness64" responseBufferLimit="0" />
For reference the following is what we replaced (the default value) that was still giving 404 errors:
<add name="ExtensionlessUrlHandler-Integrated-4.0" path="*." verb="*" type="System.Web.Handlers.TransferRequestHandler" preCondition="integratedMode,runtimeVersionv4.0" />