This is not obvious to me (on Firefox, Chrome)—how does one start off a round of kanji shiritori? In this example, the entry box prompts to start with a reading of 「す」, but not the kanji required.
There seems to be something wrong with the Kanji Shiritori round you linked to, as it is completely empty. There must be at least one word in it that you can use as a starting point. The round you linked to is also not showing up in the list of Kana/Kanji shiritoris either.
As Karlla said though, if you scroll down you get a list of all open Shiritori Kanji rounds and will certainly be able to play in those. No idea how you got to that round with no terms in it, though?
They always look like that when they are new (at least in my experience). Then (usually but not in this case) someone starts them somehow, and they show up in the list of rounds below. I've never been able to start one linked from a "New kanji shiritori round" message, so I have to wait for someone else to play.
There was an issue - when a shiritori gets started, it has no words, but the input field always states the first character to be used. In this case, though, it was set to kana by accident. Just fixed it!