In that case, you could monitor the license types in use via the Connections Overview in the DW Admin tool (we are on 6.11 so still have access to that). I am not sure where to monitor connections in other versions.
In the Connections Overview, you can expand connections and see the "Application Type". For example, we have a mix of "Platform Service" and "DocuWare Client". Platform Service is the API, and DocuWare Client is the web client.
It sounds like you would like to have this sort of monitoring be more of a log that you could report on. I am not sure what facility DocuWare has for that, but hopefully someone weighs in on that sort of audit capability. There are some logs on the server, but those appear to be mostly for error logging, not a record of every single transaction.
What version of DW are we discussing, and is it on-prem or in the cloud?