Take a look at the marketplace module:
I’m not sure about the API, but looking at this module could give some clues.
Hope this helps!
AFAIK There is no API to retrieve the logs from one Mendix application so that you can show it in another application.
As also mentioned in the above comment, you could use Logging module and the logs for the application can be shown in the same application.
The only API you have is : https://docs.mendix.com/apidocs-mxsdk/apidocs/deploy-api#3-19-download-archived-logs-for-a-specific-date-mendix-cloud-v4-only
These are deployment specific API, which helps you to download the logs.
You can also use Datadog which can be integrated with Mendix application and from the datadog portal you will be able to see the logs. But those are for more monitoring solutions. Refer : https://docs.mendix.com/howto/monitoring-troubleshooting/manage-app-performance