Yes, you can edit existing widgets. This does require the use of Developer Edition of ArcGIS Experience Builder and you need to host these widgets. The original files for the table widget can be found in the ArcGISExperianceBuilder installation folder:
It is best to load the entire ..\ArcGISExperienceBuilder\client folder into your code editor, such as VSCode.
Below are several links for videos on customizing widgets. They do focus on starting from a sample widget but provide some insight into how the widgets are assembled.
