HTTP 404 Page Not Found in Web Api hosted in IIS 7.5

I was struggling with this as well. Fortunately, Steve Michelotti documented a solution that worked for me here.

At the end of the day, I enabled all verbs (verb=”*”) to the ExtensionlessUrlHandler-Integrated-4.0 handler in my web config.

    <validation validateIntegratedModeConfiguration="false" />
    <modules runAllManagedModulesForAllRequests="true" />
        <remove name="ExtensionlessUrlHandler-Integrated-4.0" />
        <add name="ExtensionlessUrlHandler-Integrated-4.0" path="*." verb="*" type="System.Web.Handlers.TransferRequestHandler" resourceType="Unspecified" requireAccess="Script" preCondition="integratedMode,runtimeVersionv4.0" />

Others have pointed out that having WebDAV enabled causes issues. Fortunately, I did not run into that issue as well.