Begin by accessing the Amplitude API. You will need an API key and secret to authenticate your requests. Log in to your Amplitude account, navigate to the "Settings" page, and find the "Projects" section to obtain your API credentials.

Decide on the specific data you need to export. This could include event data, user properties, or cohort information. Understanding your data requirements will help you construct the correct API endpoint queries.

Use Amplitude's API documentation to construct your API request URL. For example, to retrieve event data, use the `export` endpoint (e.g., `https://amplitude.com/api/2/export`). Ensure to include necessary parameters such as start and end dates for data extraction.

Use a programming language with HTTP client capabilities (e.g., Python with `requests` module) to send the API request. Include your API credentials in the request header for authentication. Here's a basic example in Python:
```python
import requests
url = "https://amplitude.com/api/2/export"
params = {"start": "YYYYMMDDTHH", "end": "YYYYMMDDTHH"}
response = requests.get(url, auth=('API_KEY', 'SECRET'), params=params)
if response.status_code == 200:
amplitude_data = response.json()
else:
print("Error fetching data", response.status_code)
```

Once you have fetched the data, process and structure it into a format suitable for JSON. You may need to iterate through the data, filter out unnecessary fields, or transform it to match your desired structure.

Use a JSON library in your chosen programming language to write the processed data to a local JSON file. In Python, you can use the `json` module:
```python
import json
with open('amplitude_data.json', 'w') as json_file:
json.dump(amplitude_data, json_file, indent=4)
```

After writing the data to a JSON file, verify its integrity by checking for completeness and correctness. Open and inspect the JSON file to ensure it contains the expected data structure and values. You can also run a script to perform automated checks for data consistency.
Following these steps will enable you to successfully move data from Amplitude to a local JSON file without relying on third-party connectors or integrations.