The Bunny Band is a cute, cheap headgear any class can wear, handed out as part of a Kafra Corp. event. The quest is dead simple: no level or job requirement, just gather a handful of materials and bring them to a Kafra representative. Instead of buying a Bunny Band on the market, you can craft your own right at the start of the game.
Kafra Corp. Rep. in Alberta (alberta 23,232) and pick "Information" to learn the terms, then "Join" to sign up for the event.
Feather — feathers drop from many beginner-friendly flying monsters such as Picky, Condor, and Familiar.
Four Leaf Clover — a rare drop from grass and forest monsters; you can also buy one from other players.
Pearl — drops from sea and a few other monsters, or buy it on the market.
Kitty Band — the prerequisite headgear; if you don't have one yet, you can buy it or make it separately.Kafra Corp. Rep. with all the items in your inventory and talk to them — they take the materials and hand you the 
Bunny Band on the spot.