It appears Heroic Dungeons will count toward Great Vault progress in Patch 10.2.
The latest Patch 10.2 build adds some minor changes to the Great Vault requirements or rather global strings that suggest you will no longer need to complete Mythic+ dungeons to gain progress in the Dungeons category. Instead, the related global strings have been updated with the “Heroic” keyword and the “+” has been effectively removed, which means completing Heroic or Mythic Dungeons will now grant progress in the category (down from 1/4/8 Mythic+ dungeons).
We found evidence in yet another string tied to weekly rewards threshold for dungeons where the”Keystone” word was replaced with just “Mythic”.