First, visit the US Census Bureau's API documentation at https://www.census.gov/data/developers/guidance/api-user-guide.html. Familiarize yourself with the available datasets and how to construct API queries. Register for an API key if required.

Using the documentation, build a query URL for the specific dataset you are interested in. For example, if you need population data, your query might look like: `https://api.census.gov/data/2020/dec/pl?get=NAME,P1_001N&for=state:*&key=YOUR_API_KEY`. Replace `YOUR_API_KEY` with your actual API key.

Write a script in a programming language like Python to send an HTTP GET request to the constructed API URL. You can use the `requests` library in Python for this purpose. Ensure your script handles any connection errors and retries if necessary.

Once you receive the response, parse the JSON data. The response will typically be in JSON format, so you can use Python's built-in `json` module to convert the response text into a Python dictionary or list. This will allow you to manipulate the data as needed.

Depending on your specific requirements, you may need to transform the data. This could involve filtering out unnecessary fields, renaming keys, or restructuring the data for better readability or usability. Use Python's data manipulation capabilities to achieve this.

After transforming the data, write it to a local JSON file. Use Python's `json.dump()` method to serialize the data into a JSON formatted string and write it to a file on your system. Make sure to specify the appropriate file path and handle any file I/O errors.

Finally, verify the integrity and correctness of the data. Open the JSON file and manually inspect a few entries to ensure data accuracy. Additionally, you can write a small validation script to check for common data issues such as missing fields or incorrect data types.

Following these steps will enable you to move data from the US Census website to a local JSON file without the need for third-party connectors or integrations.