First, ensure you have access to the LaunchDarkly API. Log into your LaunchDarkly account and navigate to the "Account settings." Here, generate a personal access token under the "API access" section that will allow you to authenticate your requests.

Determine which data you need to export from LaunchDarkly. This could be feature flags, environments, or user segments. Make a list of the endpoints you'll need to call, as documented in the LaunchDarkly API documentation.

Write a script using a programming language of your choice (such as Python, Node.js, or Ruby) to make HTTP GET requests to the LaunchDarkly API. Use the access token generated in Step 1 to authenticate your requests. Here's a simple Python example:
```python
import requests

url = "https://app.launchdarkly.com/api/v2/flags/project-key"
headers = {
"Authorization": "Bearer YOUR_ACCESS_TOKEN"
}
response = requests.get(url, headers=headers)

if response.status_code == 200:
data = response.json()
else:
print("Failed to retrieve data:", response.status_code)
```

Once you receive the response, parse the JSON data returned by the API. Ensure to handle exceptions or errors that may occur during the parsing process. This will typically involve checking if the response is valid and extracting the necessary data fields.

Depending on your requirements, you may need to transform the data to fit your specific needs. This could involve filtering out unnecessary fields, reformatting data, or converting data structures.

With the data ready, write it to a local JSON file. Continuing with the Python example, you can use the `json` module:
```python
import json

with open('launchdarkly_data.json', 'w') as json_file:
json.dump(data, json_file, indent=4)
```

Finally, verify that the data has been correctly exported to the JSON file. Open the file and check the contents to ensure the data is formatted as expected and that all necessary information has been included.

By following these steps, you can efficiently export data from LaunchDarkly to a local JSON file without relying on third-party connectors or integrations.