To begin, you need to access the Postmark API. First, log in to your Postmark account and navigate to the 'API Tokens' section. Here, you can generate an API token if you don't have one already. This token will be necessary to authenticate your API requests.

Prepare your local development environment. Ensure you have a programming language runtime installed (e.g., Python, Node.js) that can make HTTP requests. You’ll also need a text editor or an IDE to write your script.

Depending on your chosen programming language, you may need to install libraries to handle HTTP requests. For instance, if you’re using Python, you can install the `requests` library using pip:

```bash
pip install requests
```

For Node.js, you can use the `axios` library:

```bash
npm install axios
```

Write a script to fetch data from Postmark using their API. Use the HTTP GET method to request the data you need, such as emails, messages, or related statistics. Here’s a simple example in Python:

```python
import requests

API_TOKEN = 'your_postmark_api_token'
API_URL = 'https://api.postmarkapp.com/messages/outbound'

headers = {
'X-Postmark-Server-Token': API_TOKEN,
'Accept': 'application/json'
}

response = requests.get(API_URL, headers=headers)

if response.status_code == 200:
data = response.json()
print("Data fetched successfully!")
else:
print("Failed to fetch data:", response.status_code)
```

Replace `'your_postmark_api_token'` with your actual API token.

Once you have fetched the data, process it as needed. This might include filtering specific fields or restructuring the data to fit your desired JSON format. You can manipulate the data using native data handling features in your chosen language (e.g., dictionaries in Python, objects in JavaScript).

After processing the data, write it to a local JSON file. Continuing with the Python example:

```python
import json

if response.status_code == 200:
with open('postmark_data.json', 'w') as json_file:
json.dump(data, json_file, indent=4)
print("Data written to postmark_data.json")
```

This script writes the structured data to `postmark_data.json` in the current directory.

Finally, verify the integrity and accuracy of the data written to your JSON file. Open the `postmark_data.json` file manually or use a JSON validator tool to ensure the data is correctly formatted and complete. This step helps catch any discrepancies or errors in the data extraction process.

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