As far as I know, retrieving log files is not exposed through any api or service.
However, you can subscribe to log messages via a custom java action using the java api. If you want to see an example of this, download the Logging module from the app store. The version in the app store says it is not compatible with newer Mendix version, but I have it running in a number of 6.10.x apps, you just need to upgrade a 5.x app with the module included.
This module places log messages in an entity in the domain model. From there, integration possibilities are numerous (OData, Webservices, etc.)