I'm making a DnD 5e character who specializes in the Medicine skill and has the Healer feat. Canonically, what institute of learning in Faerûn could provide the character with this training? Im looking more for an academic location verse a clerical or religious one.
Almost any Temple
Healing from clerics is common in Faerun, so most places of healing would be temples rather than schools. There are too many temples/churches to list, but some ideas may include: