Fizzbuzz for websites would be to have someone code two paragraphs side by side with a header and footer in CSS.
I suppose there are people out there that are working as front-end developers for websites that only work in HTML and CSS, but I would think that's getting more rare as time goes on. Of course, that could just be bias based on my own career.
But, going with your suggestion as a thought exercise, I would go in a different direction. To make a pattern like fizzbuzz I suppose I would do something more like: build a page with a series of paragraphs so that these CSS properties apply to every third paragraph, these CSS properties apply to every fifth paragraph, and so on. Something possible with CSS but requires the ability to think things through.
Nobody is going to color code repeating paragraphs in CSS on every project. You wouldn't use that javascript on every project either. (Perhaps these days you would I guess, but I'd argue its overkill for basic websites, to be honest) I can guarantee you're farting around with the box model though, and odds are you could be using a framework to do it for you. Which is precisely why I'd want to test it.
For what I would consider a basic website I'm doing a decent amount of CSS and JS work. But I suppose that could depend upon your definition of a basic website.