Correcting WebAPI delete/put verbs in typical ASP.NET MVC template web.config. The problem typically stems from using WebDAV, which seems to "hijack" the delete & put. You have to remove both the module (line 3) and handler (line 6).

web.config

  <system.webServer>
    <modules runAllManagedModulesForAllRequests="false">
      <remove name="WebDAVModule" />
    </modules>
    <handlers>
      <remove name="WebDAV" />
      <remove name="ExtensionlessUrlHandler-ISAPI-4.0_32bit" />
      <remove name="ExtensionlessUrlHandler-ISAPI-4.0_64bit" />
      <remove name="ExtensionlessUrlHandler-Integrated-4.0" />
      <add name="ExtensionlessUrlHandler-ISAPI-4.0_32bit" path="*." verb="GET,HEAD,POST,DEBUG,PUT,DELETE,PATCH,OPTIONS" modules="IsapiModule" scriptProcessor="%windir%\Microsoft.NET\Framework\v4.0.30319\aspnet_isapi.dll" preCondition="classicMode,runtimeVersionv4.0,bitness32" responseBufferLimit="0" />
      <add name="ExtensionlessUrlHandler-ISAPI-4.0_64bit" path="*." verb="GET,HEAD,POST,DEBUG,PUT,DELETE,PATCH,OPTIONS" modules="IsapiModule" scriptProcessor="%windir%\Microsoft.NET\Framework64\v4.0.30319\aspnet_isapi.dll" preCondition="classicMode,runtimeVersionv4.0,bitness64" responseBufferLimit="0" />
      <add name="ExtensionlessUrlHandler-Integrated-4.0" path="*." verb="GET,HEAD,POST,DEBUG,PUT,DELETE,PATCH,OPTIONS" type="System.Web.Handlers.TransferRequestHandler" preCondition="integratedMode,runtimeVersionv4.0" />
    </handlers>
  </system.webServer>