Legionfall War Supplies are a new currency after Broken Shore released in Patch 7.2. One of the core quests in Broken Shore is to used the Legionfall War Supplies to construct buildings, and unlocked some buffs, including Challenge Artifact Weapon Appearances,  Crafting materials for BOE Legendary Gears.
The player will be transfered to Redemption Outpost after finish Scenario quest in Broken Shore, then you will find some World Quests for Legionfall War Supplies from Commander Chambers.
Legionfall War Supplies are a 1,000 cap to how many you can have at any given time. Turn in 100 to one of three buildings, which can be done multiple times per day. You can earn Legionfall Recompense for completing some building quests, which rewards an Artifact Power item, some Nethershards, and Order Resources.
It's easy to do world quests to acquire Legionfall War Supplies. One quest will gain 25 Legionfall War Supplies, and 5 in Rare Elite in Broken Shore, certain amounts in Wyrmtongue Chests.