Is it possible to add a new source (to the left side) in the modern image picker dialog? A client is requesting that there be a link to an existing Image Library.
No, there is no such setting of modern image web part in SharePoint Online. To achieve it, you have to develop a web part using SharePoint Framework(SPFx). Also, I would suggest posting a uservocie here: https://feedbackportal.microsoft.com/feedback/forum/06735c62-321c-ec11-b6e7-0022481f8472. It is a place for customers provide feedback about Microsoft SharePoint products. What's more, if a feedback is high voted there by other customers, it will be promising that Microsoft Product Team will take it into consideration when designing the next version in the future.