Unlike the magic achievement, this does NOT require you to level your weapon skills up. You simply only need acquire the level 9 versions of each weapon to unlock this.
Here is a list of the orbs and where you can find them:
Sword
Initial
Mantis Ant
Pandora Treasury
Great Viper
Northtown Ruins
Chancellor of Tasnica
Mana Palace
Thunder Gigas
Mana Magic - have Primm and Popoi cast this on Randi during the final battle
Spear
Gift from Luka
Tropicallo
Pandora Treasury
Santa's House
Northtown Ruins
Gold Tower
Aegagropilon
Mech Rider 3
Rare Drop: Tsunami (Mana Fortress)
Axe
Buy: Watts
Fire Gigas
Moogle Village
Fire Palace
Emperor's Castle
Gold Tower
Dragon Worm
Rare Drop: Fiend Head (MF)
Rare Drop: Fiend Head (MF)
Boomerang
Sprite Initial
Spiky Tiger
Spring Beak
Frost Gigas
Vampire
Blue Spike
Underground City
Blue Dragon
Rare Drop: Master Ninja (MF)
Bow
Gift from Dwarf Elder
Wall Face
Jabberwocky
Boreal Face
Northtown Ruins
Gorgon Bull
Kettle Kin
Snow Dragon
Rare Drop: Metal Crab (MF)
Glove
Girl Initial
Biting Lizard
Moogle Village
Ice Palace
Metal Mantis
Dark Palace
Red Dragon
Rare Drop: Wolf Lord (MF)
Rare Drop: Wolf Lord (MF)
Whip
Gift from Witch
Scorpion Army Ship
Gemma after Jabberwocky
Mech Rider 1
Doom's Wall
Emperor's Castle
Hydra
Hexas
Rare Drop: Mace Menace (MF)
Javelin
Gift from Undine
Kilroy
Fung Castle
Minotaur
Mech Rider 2
Lime Slmie
Snap Dragon
Axe Beak
Rare Drop: Whimper (MF)