I’m still not quite sure why you’d want to shoutout with out of date information that may no longer be of any relevance (the last stream could potentially be a long time ago, or of a game they no longer play), but if that’s what you want to do and want to have that old information for every single streamer on Twitch then yes you’ll have to cache every single streamer on Twitch (and be careful of potential developer agreement issues you may run in to if you want to store this sort of data)
Perhaps you may want to use this as an opportunity to rethink your shoutout system as in most cases these sort of things can benefit from being short and to the point, rather than bloated with information users don’t really need if all they want to do is be able to follow a link to their channel.