We've known that the search functionality on the site has been busted for a while, but I received an email from an end-user that explains how to fix it (they also use DocFx):

<?xml version="1.0"?>

<configuration>

    <system.webServer>

        <staticContent>

            <mimeMap fileExtension=".json" mimeType="application/json" />

                </staticContent>

    </system.webServer>

</configuration>

Since we're using Azure Websites to host the docs, which IIRC is running on IIS, we should be able to slap that configuration at the root of the website and fix it that way.