Can I download published videos from twitch and use them on the soundboard?

It says ^^ above

No you are not allowed.

The copyright lies with the publisher.

Now I'm not sure where the Twitch servers are, but the corresponding national law applies.

At least you must of course specify the source and the author of the video.

It's best to read the Twitch Terms of Use again to be on the safe side before any problems arise.

PS: alternatively, on Twitch you can also contact the channel operator of the content you want to use and ask for their permission. If he then agrees, the whole thing is legitimate.