Active Administrator does not have any integration with third party Syslog servers/systems such as Splunk, nor do we have a defined or supported method of these types of systems obtaining data from the AA database.
You may be able to query the AA database to return records to the Syslog server. The table that stores the AA captured events is dbo.tblAlertLog. You will most likely want to have your query join with the dbo.tblEventDefinition table to match the EventDefinitionCode so you can get the DefinitionName and possibly the EventSource. Joining with the dbo.tblComputer to match the ComputerID with the ComputerName will identify which machine the event came from.
Please note, creating a query to return the data is beyond the scope of Support.