From now on for any mission you do in the Mission Vault (which excludes Master Missions, Badge Runs, and Island Trials) you have a high chance of receiving a Snag Emblem Piece (so long as you actually put effort into the mission). You can only receive a single Snag Emblem piece for each individual mission, once. Three Snag Emblem Pieces can be used in place of a single Snag Emblem anywhere, including alchemy.

Note: Exceptional missions can still earn you a whole Snag Emblem on top of a Snag Emblem Piece.