I want to stream Pokemon White on Twitch using an emulator. Is this allowed? I bought the game normally back then!
I see nothing that speaks against emulators; also nothing that speaks against it, as you plan it.
No emulator can be banned for, even if you bought the game as original then you have to download it for the emulator as ROM and this is illegal.
Of course there's something against this, namely that it is illegal to download ROMs for emulators free of charge.
The question is not "free"
In theory such a rom can always be drawn by yourself and that would even be legal. In practice, however, nobody does it, but simply downloads them illegally from the Internet.
In theory, the Rom can itself be drawn from the original, which would be legal. But of course nobody does it in practice.