You can use this opensource service to at least read other people's tweets:
https://nitter.net/ https://nitter.net/about
See an example: https://nitter.net/elonmusk
https://nitter.net/ https://nitter.net/about
See an example: https://nitter.net/elonmusk
Without API and without web access, where are you going to read the tweets from?
The only option left is unofficial API stubs, which essentially scrape the web requests, but those also will require a login, so the high activity will be easy to spot and block by Twitter.
Unless they have a whole network of accounts. That'd work but it's work to do.
edit: Well, worked. Seems down right now, possibly because it was the only instance that still worked.