Start by cloning the GitHub repository to your local machine. Open your terminal or command prompt, navigate to the directory where you want to clone the repository, and run the command `git clone `. This will create a local copy of the repository on your machine.

Once the repository is cloned, navigate into the directory using the command `cd `. This step ensures that you are working within the correct context of your project files.

Locate the specific file(s) in the cloned repository that contains the data you want to convert to JSON. These files could be in various formats like CSV, XML, or even another JSON format that needs restructuring.

Use a programming language such as Python to read the data file. For instance, if the file is a CSV, you can use Python’s built-in `csv` module or the `pandas` library for reading. Use a script similar to this:
```python
import pandas as pd

data = pd.read_csv('datafile.csv') # Adjust the filename and path as needed
```

Once the data is loaded into your program, convert it to a JSON format. In Python, you can easily do this using the `to_json()` method if you are using pandas, or `json` module for other data types:
```python
json_data = data.to_json(orient='records') # For pandas DataFrame
```
Or, if you’re using the `json` module for a dictionary:
```python
import json

json_data = json.dumps(data_dict) # Assuming data_dict is your data structure
```

Write the JSON data to a local file. Ensure you specify the correct file path and name for where you want to save this data:
```python
with open('local_data.json', 'w') as json_file:
json_file.write(json_data)
```

Finally, verify the contents of the newly created JSON file to ensure the data is correctly structured and matches the original data from the GitHub repository. You can do this by opening the file in a text editor or by loading it back into your programming environment and inspecting it.

By following these steps, you will successfully move data from a GitHub repository to a local JSON file without using third-party connectors or integrations.