I can confirm that turning in "Supplies" quests (bonus chest from +10k rep with an exalted Legion faction) does not give you any essences. :-( Perhaps it was this way for the other legendary item upgrade quests, but I wasn't aware, so thought it would be good to share the info.
To consolidate info from another wowhead post, here is where you CAN get essences:
- Daily emissary cache (4-10 per cache)
- Daily random heroic dungeon finder (5 daily)
- Antorus boss kills - only receive essences once per boss, regardless of difficulty (i.e. killing the first boss on lfr, normal, and heroic, you still only get essences 1 of those times) (3-6 per boss)
- Mythic + cache (20 essences, regardless of mythic difficulty completed the prior week)
- Weekly PvP quest rewards (i.e. earning 1500 honor in 2s, 3s, and rated BGs) (7 each quest completion)
source: http://www.wowhead.com/news=277232/legendary-upgrade-clarifications-in-antorus?webhook/legendary-upgrade-clarifications-in-antorus