Did you know you can also use a data-URI as a favicon? And if you use a <text> element inside that SVG, you can set any text as a favicon - including an emoji.

For example, this sets the favicon to a little picture of the text "!!!":

    <link
    rel="icon"
    href='data:image/svg+xml,<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 100 100"><text y=".9em" font-size="90">!!!</text></svg>'
    />

You can also replace the "!!!" in the above snippet with any emoji, and it works. (I couldn't do an emoji here because HN filters them out.)

You could maybe add this as an option for copying code from your site, and the result will work without any request to any server (with the caveat that the emoji design will vary depending on OS).