I am not sure why you would want to access the logs at Runtime. When an exception occurs you can actually handle the exception directly. Logs should be used to triage a problem so that you can correct coding issues and add more exception handling to the code.
If you want users to send you logs when they have an issue, you can programatically check the CommonConfig,xml file for the location where files are being written - it is the ConfigurationLocation key that specifies that. You could create a mechanism for them to report an issue which would send the current log file. Best to zip the file if you are emailing it as they can get rather large over time. Check out this post with some examples for zipping a file using components installed with the Runtime.
Logs are found by default in %appdata%\Roaming\OpenSpan if logging is enabled. Any user should have access to that location. If RuntimeLog logging is enabled there is a single file for the entire session (RuntimeLog.txt) - if Log4Net logging is enabled rolling logs will exist and logging for the session may be split between many files (Runtime_Log4Net.txt*). Here is a post that will help you get to the important data in a log quickly.