The NASA tool that allows users to track what the Hubble and Webb satellites are currently observing, and to browse and request observations as well, is the JWST Observing Explorer.

* For HST: Go to https://hst-ao.stsci.edu/scheduling/obslist.php

* For JWST: Go to https://jwst-obslist.stsci.edu/public_explorer/browse_requests

Each observation is tagged with its target name to aid your search to see any specific information available.