https://www.thetrumparchive.com/faq

Can I have the data?

Yes. You can download the full dataset via Google Drive either as a CSV file or JSON file (last updated 11/06/2020). You can credit the site however you'd like. Note that the dates are stored in UTC time.

There's also an API endpoint to pull the latest 1000 tweets: https://www.thetrumparchive.com/latest-tweets (cached for 30 minutes).