Right now, someone ended a Kanji Shiritori in 初代, so I thought I can now use 代々木, which is a well-known place and does not end in 々, as it says in the rules.
However, if I want to play it, I'm being told that I can only use words made out of Kanji only. Is this maybe an issue with words using 々?
You can't use 々 anywhere. If you want to use a word with a repeating kanji, you need to write the kanji in full instead of using 々. I'm fairly certain it's not a bug as it has been like this for years.