If you are in the hole where you really cannot add load to your database server but want to log the queries, there is a technique called zero impact monitoring

If you are in the hole where you really cannot add load to your database server but want to log the queries, there is a technique called zero impact monitoring where you literally mirror the network traffic going to your database server, and use a separate server to reconstruct it into query logs. These logs identify the queries that are being run, and critically, who/what is running them. A past workplace of mine has used this approach.