Unfortunately, the functionality you are looking for doesn't exist in the web data connector platform. There isn't currently any way to change the extract data dynamically based on user. This is a great feature request though. I would encourage you to make a post in the Ideas forum (this is where other users can vote on features that they also need). This helps us figure out the common pain points of the community.
Right now, the only workaround is to have one data source per user. For example, each of your users could create a web data connector with their own credentials, and they would then get a custom extract that they could interact with and publish up to Server. You could of course create the data sources for them as well.
Let me know if you have any questions about this approach. And thank you for the feedback! It is very appreciated and helpful.
Thanks Brendan, appreciate the response and feedback.
The workaround that you suggested will not scale with our client base.