Before you can access Mailgun data, you need to set up API credentials. Log into your Mailgun account, navigate to the "API Keys" section, and note down the "Private API Key". You'll use this key to authenticate your requests.

Familiarize yourself with the Mailgun API documentation (https://documentation.mailgun.com/). Identify the endpoints that provide the data you want to export. Common endpoints include messages, logs, and stats.

Make sure you have a command-line tool like `curl` or a programming language environment like Python or Node.js installed on your local machine. These tools will allow you to send HTTP requests to the Mailgun API.

Write a script in your preferred language to fetch data from Mailgun. For example, using Python, you can use the `requests` library to send a GET request to the desired Mailgun API endpoint. Include your API credentials in the request headers for authentication.

```python
import requests

def fetch_mailgun_data():
url = "https://api.mailgun.net/v3/YOUR_DOMAIN_NAME/messages"
auth = ("api", "YOUR_API_KEY")
response = requests.get(url, auth=auth)
return response.json()
```

Once you have the data from Mailgun, parse it to extract relevant fields. You might need to loop through the response to collect specific information. Format this data into a JSON-compatible format if it isn't already.

Use file-handling operations to write the parsed data to a local JSON file. In Python, you can use the `json` library to handle this.

```python
import json

def save_to_json(data, filename='mailgun_data.json'):
with open(filename, 'w') as file:
json.dump(data, file, indent=4)
```

If you need to move data regularly, consider setting up a cron job (on Unix-based systems) or Task Scheduler (on Windows) to run your script at specified intervals. This will automate data fetching and storage without manual intervention.

By following these steps, you can efficiently move data from Mailgun to a local JSON file without relying on third-party connectors or integrations.