There’s the GitHub - discourse/discourse-poll-banner: Display a poll from a topic as a banner for all logged-in users. · GitHub theme component that works off a numerical poll. I don’t know if someone could adapt it to handle ‘word-based’ ones in some way?
There does seem to be a scattering of topics about having polls in banners, but I think a lot of them were initially bug reports that have been reclassified rather than a proper Contribute > Feature request. It may be good to try and collate something to give it a better hearing and allow people to add their voice to the conversation.