Move chat section next to video

Yes, you can do this easily using CSS.
Video and chat are output in two divs:

<div class="lsb-embedded-view">...</div>
<div class="lsb-embedded-chat">...</div>

You will need to edit your theme’s style.css and add styling for them.

I know this is a bit old but would it at all be possible to have this functionality by default? I don’t want to come off sounding incredibly lazy but it is kind of a hassle to have to go in and manually edit things.

If not I understand, but otherwise I don’t see the value of using a plugin for this when embedding the twitch stream and chat with the regular codes seems to work fine. I guess I just don’t see the advantage of using a plugin if it doesn’t make things much easier.