A couple of things you can try here, all web.config related, firstly modify your modules element to include the attribute runAllManagedModulesForAllRequests="true"
, as below:
<modules runAllManagedModulesForAllRequests="true">
<remove name="WebDavModule" />
</modules>
Then set your handlers to the below:
<handlers>
<remove name="ExtensionlessUrlHandler-ISAPI-4.0_32bit" />
<remove name="ExtensionlessUrlHandler-ISAPI-4.0_64bit" />
<remove name="ExtensionlessUrlHandler-Integrated-4.0" />
<remove name="WebDav" />
<remove name="OPTIONSVerbHandler" />
<add name="ExtensionlessUrlHandler-ISAPI-4.0_32bit" path="*." verb="GET,HEAD,POST,DEBUG,PUT,DELETE,PATCH,OPTIONS" modules="IsapiModule" scriptProcessor="%windir%Microsoft.NETFrameworkv4.0.30319aspnet_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.NETFramework64v4.0.30319aspnet_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>
This should do the trick, but if it doesn't, as a last resort you can force IIS to output the correct headers with the below:
<system.webServer>
<httpProtocol>
<customHeaders>
<add name="Access-Control-Allow-Origin" value="*" />
<add name="Access-Control-Allow-Methods" value="GET,PUT,POST,DELETE,OPTIONS" />
<add name="Access-Control-Allow-Headers" value="Content-Type" />
</customHeaders>
</httpProtocol>
</system.webServer>
Be wary of the wildcard value, you should really set this to the domain name that your site will be hosted on.