Can StreamAlive be embedded?

Yes, StreamAlive absolutely can be embedded (including into an Learning Management System or LMS). We use the In-person Event functionality where we provide the chat app that can be embedded into a website using an IFRAME approach. The underlying technology has been used on YouTube Livestreams with thousands of concurrent users.

it's as simple as adding this code to a web page:
<iframe src="https://chat.streamalive.com" title="StreamAlive Browser Chat"></iframe>

You would have to make some edits, like adding the width that you want for the iframe.