First, you need to access the Greenhouse API. Visit the Greenhouse Developer portal and log in with your credentials. Navigate to the API documentation to find the available endpoints you can use to extract data, such as job listings, candidates, or applications. Ensure you have the necessary permissions and API key to access the data you need.

Set up your local development environment to make HTTP requests. You can use a programming language like Python, JavaScript (Node.js), or another language that supports HTTP requests. Install any necessary libraries or frameworks. For example, in Python, you can use the `requests` library to handle HTTP requests.

Use your API key to authenticate your requests. Typically, you will include the API key in the headers of your HTTP requests. For instance, in Python, you might set up your headers like this: `headers = {'Authorization': 'Bearer YOUR_API_KEY'}`. This step is crucial to ensure you have access to the data and that your requests are authorized.

Construct and send GET requests to the Greenhouse API endpoints relevant to the data you wish to extract. For example, if you are extracting job postings, you might send a GET request to `https://harvest.greenhouse.io/v1/jobs`. Use your programming language's HTTP request library to send these requests and handle the responses.

Once you receive the response from the API, parse the JSON data. Most programming languages have built-in or easily accessible libraries to handle JSON data. For instance, in Python, you can use the `json` module to parse the response: `data = response.json()`. Ensure that you handle any potential errors or exceptions that may occur during parsing.

Organize the parsed data into the structure you want to save locally. This might involve selecting specific fields, renaming keys, or nesting data. Create a Python dictionary or a JavaScript object that represents the structure you want for your local JSON file.

Finally, write the structured data to a local JSON file. Use your language's file handling capabilities to open (or create) a file and write the JSON data. For example, in Python, you can use:
```python
with open('data.json', 'w') as f:
json.dump(data, f, indent=4)
```
This will create a file named `data.json` with your extracted data formatted in JSON. Make sure to handle file I/O exceptions to ensure the file is written correctly.

This guide provides a fundamental approach to moving data from Greenhouse to a local JSON file without using third-party connectors or integrations. Adjust the steps as necessary to fit the specific requirements of your data and environment.