A Twilio account is required to build a SMS Haiku Generator with the help of the Datamuse API and the Syllable module. The process involves creating a new project using the Twilio Serverless Toolkit, installing necessary dependencies, writing a function in Node.js that generates a haiku according to an input text message, and configuring the Twilio Function with a public-facing URL for deployment. The complete code is available on GitHub, and users can explore various possibilities with the Twilio Serverless Toolkit.