Begin by familiarizing yourself with the Recharge API documentation. This will provide you with the necessary endpoints and authentication requirements. Recharge's API allows you to access customer, order, and subscription data, among others, which is crucial for extracting the data you need.

To interact with the Recharge API, you need to authenticate your requests. Typically, this involves using an API key. You can obtain your API key from the Recharge admin dashboard under the API settings. Make sure to securely store this key as it is sensitive information.

Decide on a programming language you're comfortable with to handle HTTP requests and JSON data. Popular choices include Python, JavaScript (Node.js), or Ruby. Each of these languages has libraries that can simplify making HTTP requests and handling JSON data.

Develop a script using your chosen language to make HTTP GET requests to the Recharge API endpoints. For example, if you want to fetch customer data, you would send a request to the "/customers" endpoint. Ensure your requests include necessary headers, such as authentication tokens.

```python
import requests

API_KEY = 'your_api_key'
headers = {
'X-Recharge-Access-Token': API_KEY
}
response = requests.get('https://api.rechargeapps.com/customers', headers=headers)
data = response.json()
```

After retrieving the data, process it as needed. This might involve filtering out unnecessary information or restructuring the data to fit your needs. Ensure that the data is in the correct format for JSON serialization.

Once the data is processed, write it to a local JSON file. This involves serializing your data into JSON format and writing it to a file on your system. In Python, you can use the `json` module for this purpose.

```python
import json

with open('data.json', 'w') as json_file:
json.dump(data, json_file, indent=4)
```

If you need to regularly update your local JSON file with new data from Recharge, consider automating the process. You can use cron jobs on Unix-based systems or Task Scheduler on Windows to run your script at specified intervals.

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