This is the order for the weapons. Bold means it's already acquired. The weapon in brackets means the person is next for the said weapon.
Celi Lucis → High Allagan Crook (High Allagan Grimoire of Healing)
Electro Witch → High Allagan Battleaxe (High Allagan Baghnakhs)
Inkin Tamusi → High Allagan Composite Bow (High Allagan Crook)
Jaqen H'ghar → High Allagan Staff (High Allagan Spear)
Lil Scorpy → High Allagan Grimoire of Casting (High Allagan Crook)
Luna Vespertine → High Allagan Grimoire of Healing (High Allagan Composite Bow)
Ricardo Ramos → High Allagan Baghnakhs (High Allagan Composite Bow)
Tapko Wild → High Allagan Blade (High Allagan Battleaxe)