Unless Supabase decides to integrate Lantern (currently they integrate pgvector) you unfortunately cannot use Lantern with Supabase.
That said, we will offer Lantern Cloud, our own hosted postgres offering (very soon. Happy to keep you in the loop. If you’re interested, please feel free to join the waitlist here: https://forms.gle/PouJxAWiSa63udJW8
Not a Supabase user, but to my knowledge -- no. Except for a set of blessed extentions, Supabase only supports extensions that are written in "trusted languages" (ie. supported by pg_tle), and Lantern is written in C, which is currently not supported.
