UPDATE: I've since received data-mined information and constructed a complete guide here: https://www.reddit.com/r/zelda/comments/5y3r8o

Here is a list of items I've documented coming from various amiibo in Breath of the Wild. I savescummed to do lots and lots of scans, but I'm pretty sure there are a few more items still missing. The RNG can be pretty mean, sometimes taking several weeks' worth of scans to give you a specific item.

How amiibo work in BotW: You can scan a specific amiibo once per day. The day resets at midnight (according to your system clock), and reloading a save from before you scanned will also reload the day's scan (allowing you to savescum to get something good). A scan will give you a drop of some random supply items and a chest containing one valuable item.

The way this list works: "Chest" denotes an item that can come from the main chest that drops. "Drop" denotes what that amiibo gives you besides the chest. You only get one chest per scan (except for Rider Link, which will give you two chests at the same time containing the Traveler's Bridle and Saddle). Parentheses after a weapon name denote stat bonuses. Epona is a special case; instead of any items dropping from the sky, Epona will appear on the ground in a flash of light and be ready to mount.

Wolf Link is also compatible with this game, but doesn't give any items. Instead, it will bring Wolf Link into BotW to fight alongside you as a companion. Once Wolf Link dies, he cannot be brought back until the next real-life day. If he simply gets lost (such as when you enter and exit a Shrine), you are allowed to re-scan and summon him back. His health will be higher depending on your saved performance from the Cave of Shadows in Twilight Princess HD.

All non-Zelda-series amiibo are also compatible, but only give a few random common materials when scanned.

This list is still being updated! Don't hesitate to report anything you don't see on this list to me on Reddit: /u/ShugoTakahashi

Super Smash Bros. series

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Link

-Epona (The legendary horse! Max bond, 4 stars in all stats, gentle temperament, and her white hair glows in the dark.)

-Chest: Cap of Twilight

-Chest: Tunic of Twilight

-Chest: Trousers of Twilight

-Chest: Soldier's Broadsword (Durability Up)

-Chest: Soldier's Broadsword (Critical Hit)

-Chest: Soldier's Shield (Durability Up)

-Chest: Fire Arrow x5

-Chest: Ice Arrow x5

-Chest: Shock Arrow x5

-Chest: Bomb Arrow x5

-Drop: random fruit

Zelda

-Chest: Twilight Bow (Attack Up + 7)

-Chest: Royal Bow (Attack Up + 8)

-Chest: Knight's Bow (Attack Up + 6)

-Chest: Knight's Bow (Durability Up)

-Chest: Soldier's Bow (Attack Up + 6)

-Chest: Soldier's Bow (Durability Up)

-Chest: Star Fragment

-Chest: Diamond

-Chest: Sapphire

-Chest: Ruby

-Chest: Topaz

-Chest: Luminous Stone

-Chest: Opal

-Chest: Amber

-Drop: random herbs/vegetables

Sheik

-Chest: Sheik's Mask

-Chest: Eightfold Longblade (Critical Hit)

-Chest: Eightfold Longblade (Durability Up)

-Chest: Eightfold Blade (Durability Up)

-Chest: Shield of the Mind's Eye (Durability Up)

-Chest: Fire Arrow x5

-Chest: Ice Arrow x5

-Chest: Shock Arrow x5

-Chest: Bomb Arrow x5

-Drop: random mushrooms

Ganondorf

-Chest: Sword of the Six Sages (Durability Up)

-Chest: Golden Claymore (Attack Up + 8)

-Chest: Golden Claymore (Critical Hit)

-Chest: Knight's Claymore (Attack Up + 7)

-Chest: Knight's Claymore (Critical Hit)

-Chest: Soldier's Claymore (Attack Up + 9)

-Chest: Soldier's Claymore (Durability Up)

-Chest: Star Fragment

-Chest: Diamond

-Chest: Sapphire

-Chest: Ruby

-Chest: Topaz

-Chest: Luminous Stone

-Chest: Opal

-Chest: Amber

-Chest: Moblin Guts

-Chest: Lizalfos Tail

-Drop: barrels containing arrows and rupees

Toon Link

-Chest: Cap of the Wind

-Chest: Tunic of the Wind

-Chest: Trousers of the Wind

-Chest: Sea-Breeze Boomerang (Durability Up)

-Chest: Knight's Broadsword (Attack Up + 5)

-Chest: Soldier's Broadsword (Attack Up + 6)

-Chest: Soldier's Broadsword (Durability Up)

-Chest: Soldier's Broadsword (Critical Hit)

-Chest: Fire Arrow x5

-Chest: Ice Arrow x5

-Chest: Shock Arrow x5

-Chest: Bomb Arrow x5

-Drop: random fish

30th Anniversary - The Legend of Zelda series

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Link (The Legend of Zelda)

-Chest: Cap of the Hero

-Chest: Tunic of the Hero

-Chest: Trousers of the Hero

-Chest: Sword (Critical Hit)

-Chest: Knight's Broadsword (Critical Hit)

-Chest: Soldier's Broadsword (Attack Up + 6)

-Chest: Soldier's Broadsword (Durability Up)

-Chest: Knight's Shield (Durability Up)

-Chest: Soldier's Shield (Durability Up)

-Chest: Fire Arrow x5

-Chest: Ice Arrow x5

-Chest: Shock Arrow x5

-Chest: Bomb Arrow x5

-Drop: barrels containing arrows and rupees

Link (Ocarina of Time)

-Chest: Cap of Time

-Chest: Tunic of Time

-Chest: Trousers of Time

-Chest: Biggoron's Sword (Attack Up + 10)

-Chest: Biggoron's Sword (Durability Up)

-Chest: Royal Claymore (Attack Up + 20)

-Chest: Knight's Claymore (Attack Up + 7)

-Chest: Soldier's Claymore (Attack Up + 9)

-Chest: Soldier's Claymore (Critical Hit)

-Chest: Fire Arrow x5

-Chest: Ice Arrow x5

-Chest: Shock Arrow x5

-Chest: Bomb Arrow x5

-Drop: random raw meat

Toon Link (The Wind Waker)

-Chest: Cap of the Wind

-Chest: Tunic of the Wind

-Chest: Trousers of the Wind

-Chest: Sea-Breeze Boomerang (Attack Up + 7)

-Chest: Knight's Broadsword (Attack Up + 5)

-Chest: Soldier's Broadsword (Durability Up)

-Chest: Boomerang (Attack Up + 8)

-Chest: Boomerang (Critical Hit)

-Chest: Boomerang (Durability Up)

-Chest: Fire Arrow x5

-Chest: Ice Arrow x5

-Chest: Shock Arrow x5

-Chest: Bomb Arrow x5

-Drop: random fish

Toon Zelda (The Wind Waker)

-Chest: Hero's Shield (Durability Up)

-Chest: Royal Shield (Durability Up)

-Chest: Knight's Shield (Durability Up)

-Chest: Soldier's Shield (Durability Up)

-Chest: Star Fragment

-Chest: Diamond

-Chest: Sapphire

-Chest: Ruby

-Chest: Topaz

-Chest: Luminous Stone

-Chest: Opal

-Chest: Amber

-Drop: random herbs/vegetables

The Legend of Zelda: Breath of the Wild series

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Link (Archer)

-Chest: Royal Bow (Attack Up + 8)

-Chest: Royal Bow (Durability Up)

-Chest: Knight's Bow (Attack Up + 6)

-Chest: Knight's Bow (Durability Up)

-Chest: Soldier's Bow (Attack Up + 6)

-Chest: Soldier's Bow (Durability Up)

-Chest: Fire Arrow x5

-Chest: Ice Arrow x5

-Chest: Shock Arrow x5

-Chest: Bomb Arrow x5

-Drop: random raw meat and fish

Link (Rider)

-Chest: Traveler's Bridle

-Chest: Traveler's Saddle

-Chest: Royal Broadsword (Durability Up)

-Chest: Soldier's Broadsword (Attack Up + 6)

-Chest: Soldier's Broadsword (Critical Hit)

-Chest: Soldier's Broadsword (Durability Up)

-Chest: Fire Arrow x5

-Chest: Ice Arrow x5

-Chest: Shock Arrow x5

-Chest: Bomb Arrow x5

-Drop: random mushrooms

Zelda

-Chest: Royal Shield (Durability Up)

-Chest: Knight's Shield (Durability Up)

-Chest: Soldier's Shield (Durability Up)

-Chest: Star Fragment

-Chest: Diamond

-Chest: Sapphire

-Chest: Ruby

-Chest: Topaz

-Chest: Luminous Stone

-Chest: Opal

-Chest: Amber

-Drop: random herbs and vegetables

Guardian

-Chest: Guardian Sword++ (Durability Up)

-Chest: Guardian Sword+ (Critical Hit)

-Chest: Guardian Spear+ (Critical Hit)

-Chest: Guardian Spear+ (Durability Up)

-Chest: Guardian Spear (Critical Hit)

-Chest: Guardian Spear (Durability Up)

-Chest: Ancient Battle Axe (Critical Hit)

-Chest: Guardian Shield++ (Durability Up)

-Chest: Guardian Shield+ (Durability Up)

-Chest: Ancient Core

-Chest: Ancient Arrow x3

-Drop: iron crates containing ancient machine parts, rare gems, and arrows

Bokoblin

-Chest: Dragonbone Boko Club (Critical Hit)

-Chest: Dragonbone Boko Club (Long Throw)

-Chest: Dragonbone Boko Club (Durability Up)

-Chest: Dragonbone Boko Bat (Critical Hit)

-Chest: Dragonbone Boko Spear (Critical Hit)

-Chest: Dragonbone Boko Spear (Durability Up)

-Chest: Spiked Boko Club (Critical Hit)

-Chest: Spiked Boko Club (Durability Up)

-Chest: Spiked Boko Bat (Critical Hit)

-Chest: Spiked Boko Spear (Durability Up)

-Chest: Dragon Bone Boko Bow (Attack Up + 8)

-Chest: Dragonbone Boko Shield (Durability Up)

-Chest: Spiked Boko Shield (Durability Up)