I think your best road would be to create Mendix filedocuments. This way the user does not even have to leave his app to retrieve the logs. You will find some posts in the forum on how to create Mendix filedocuments in Java actions.
The logs can be found in your sprintr environment. But depending on what you are trying to achieve you could also look at this app: https://appstore.home.mendix.com/link/app/159/eMagiz/Logging-Module
You can use the logger provided by Mendix: Core.getLogger(...).info()