If an adventurer's HP falls to zero, they will not gain any medals. If you use any revives or continues, none of your adventurers will gain any medals.
When playing co-op, only adventurers you yourself control will be eligible for medals. Also, if a player disconnects immediately after a quest begins and your own adventurer(s) are forced to take their place, those adventurers will not be able to obtain any medals.
Adventurers that are only temporarily part of your roster can only earn medals once they have permanently joined you.