Before you begin, you need to understand the CallRail API and its documentation. Visit the official CallRail API documentation to identify the endpoints you'll need to access the data you want to export.

1. Log in to your CallRail account.
2. Navigate to the Account Settings and then to the integrations or API section.
3. Generate an API key. This key will be used to authenticate your requests to the CallRail API.

Decide on the data you want to retrieve from CallRail. This could be call logs, text messages, form submissions, etc. Identify the corresponding API endpoints for the data.

Using a programming language of your choice (e.g., Python, JavaScript, or Ruby), write a script that makes an HTTP request to the CallRail API endpoints.

Here's an example in Python using the `requests` library:

```python
import requests
import json

# Replace 'your_api_key' with your actual CallRail API key
api_key = 'your_api_key'
headers = {
   'Authorization': f'Token token="{api_key}"',
   'Content-Type': 'application/json'
}

# Replace with the correct API endpoint for the data you want to retrieve
endpoint = 'https://api.callrail.com/v3/calls.json'

response = requests.get(endpoint, headers=headers)

if response.status_code == 200:
   data = response.json()
   # Do something with the data
else:
   print(f'Error: {response.status_code}')
```

‍

After fetching the data from the API, you'll likely have it in a JSON format already. If not, or if you need to transform the data, you can use the `json` module in Python to serialize the data into a JSON string.

```python
# Assuming 'data' is a Python dictionary containing the API response
json_data = json.dumps(data, indent=4)

# To save the JSON data to a file
with open('callrail_data.json', 'w') as file:
   file.write(json_data)
```

Run your script to ensure it works correctly. Check the JSON output file to verify that the data is structured as expected.

Add proper error handling to your script to manage any potential issues, such as network problems, API rate limits, or unexpected data formats.

```python
try:
   response = requests.get(endpoint, headers=headers)
   response.raise_for_status()  # Raises an HTTPError if the HTTP request returned an unsuccessful status code
   data = response.json()
except requests.exceptions.HTTPError as errh:
   print(f'Http Error: {errh}')
except requests.exceptions.ConnectionError as errc:
   print(f'Error Connecting: {errc}')
except requests.exceptions.Timeout as errt:
   print(f'Timeout Error: {errt}')
except requests.exceptions.RequestException as err:
   print(f'Error: {err}')
```

‍

If you need to move data from CallRail to JSON regularly, consider scheduling your script to run at specific intervals using a task scheduler like `cron` on Linux or Task Scheduler on Windows.

Ensure that your API key and the JSON data file are stored securely. Avoid hardcoding your API key directly into your script. Instead, use environment variables or a configuration file that is not included in your version control system.

Finally, review your script's performance and output. Refine as needed to ensure it meets your requirements for data structure and integrity.

By following these steps, you should be able to move data from CallRail to JSON format without the need for third-party connectors or integrations. Remember to adhere to best practices for security and error handling.