Begin by reviewing the Mailjet SMS API documentation to understand the structure, authentication, and endpoints. Familiarize yourself with the necessary HTTP requests to retrieve SMS data. You'll need your Mailjet API key and secret to access the API.

Choose a programming language you're comfortable with (e.g., Python, JavaScript) and set up your development environment. Ensure you have the necessary libraries to make HTTP requests and handle JSON data, such as `requests` for Python or `axios` for JavaScript.

Write a script to authenticate with the Mailjet API using your API key and secret. Construct an HTTP GET request to the appropriate Mailjet SMS endpoint to retrieve the desired data. Ensure you handle any necessary query parameters to filter or limit the data returned.

Once you receive a response from the Mailjet API, parse the JSON data. Extract the relevant fields that you need to store locally. Typically, you'll use a JSON parsing library or function available in your chosen programming language.

After parsing, format the extracted data into a JSON structure suitable for local storage. Ensure the data is clean and correctly structured, adjusting field names or formats if necessary to meet your specific requirements.

Use file handling functions in your programming language to write the formatted data to a local JSON file. Open a file in write mode and output the JSON data, ensuring the file is correctly closed after writing to avoid data corruption.

To regularly update your local JSON file with new SMS data, automate this script. You can use cron jobs in Unix-based systems or Task Scheduler in Windows to run the script at desired intervals, ensuring your local data remains current.