Begin by visiting the Customer.io API documentation page. Understand the available endpoints and authentication method. You will need to use the API to extract data, typically through RESTful requests. Note any necessary API keys or tokens required for authentication.

Log in to your Customer.io account and navigate to the API settings. Generate an API key if you haven't already. Ensure you have the correct permissions to access the data you need. Keep your API credentials secure as they provide access to your Customer.io account.

Ensure you have tools like `curl` or a programming language environment (such as Python with the `requests` library) installed on your local machine. These tools will allow you to make HTTP requests to the Customer.io API.

Construct API requests to retrieve the desired data from Customer.io. For instance, using `curl`, you might use a command like:
```bash
curl -X GET "https://api.customer.io/v1/api/customers" -u YOUR_SITE_ID:YOUR_API_KEY
```
If using Python, a request might look like:
```python
import requests

response = requests.get(
"https://api.customer.io/v1/api/customers",
auth=('YOUR_SITE_ID', 'YOUR_API_KEY')
)
data = response.json()
```

Ensure the data retrieved from Customer.io is in JSON format. If using `curl`, you might redirect the output to a JSON file. In Python, ensure you convert the response to JSON:
```python
import json

with open('data.json', 'w') as json_file:
json.dump(data, json_file, indent=4)
```
This will save the data to a `data.json` file on your local machine.

Open the JSON file to verify that the data has been saved correctly. Check for any discrepancies or issues in the data. If necessary, clean up any unwanted fields or errors in the JSON file using a text editor or programmatically.

If you need to regularly update the local JSON file with data from Customer.io, consider writing a script (e.g., in Python) that automates the above steps. You can schedule this script to run at regular intervals using cron jobs (on Unix-like systems) or Task Scheduler (on Windows).

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