To use this function you need to register (for free) at USGS.
Go to https://earthexplorer.usgs.gov/ and log in.
Now draw a bounding box of your desired area or upload a shapfile or KML. Note that this is limited to 30 vertex:
Add additional criteria such as a time range or the number of visible results (you can download unlimited, see below) if you want to.
Now select Sentinel 2:
The results will be displayed. You can export your result as .CSV or other formats:
Make sure to choose "Non limited results" before exporting if you want to download all results without any limit:
After a few minutes an email will be sent to you including a link to this file. It looks like this for example: