And if so, are pure listening bots that never speak also prohibited? These could be useful for custom notification.
Chat bots (that post messages) have no place in TNB. Since TNB is for discussion of code-golf and PPCG related topics, having bots in the room would be distracting and obnoxious, and would overtake more important conversations. The reason that this is not officially mentioned in the chatiquette is because this isn't a frequent or reoccurring issue. You could try posting an answer here if you think this warrants being added to the official TNB guidelines.
Now, bots can be super helpful and useful at times. So if you have a bot and you would like to run it in a chat room, that's perfectly fine as long as it goes in the appropriate room. Testing the bot should go in the sandbox, and once the bot works correctly, it should go in it's own new room or in an existing room with the permission of that room's owner.
Bots that don't post any messages are an exception. If they are only listening to the ongoing conversation, they can't derail the conversation or annoy users, so there is no harm from having them in TNB. Of course, while testing that the bot is functioning correctly, this should also go in the sandbox.