Just to be sure I understand correctly the issue. The problems you have is creating and serving the shiny app, and then authenticating to Twitter's API?
I'm not sure what is the purpose of scraping Twitter data, check if it is according to the Twitter API policies.
You also do not show, what version of rtweet are you using or how do you authenticate. You seem to be using the user authentication. You should probably use the consumer_key, consumer_secret, access_token , access_secret of your own app to avoid using the browser/user authentication.