The Wizard's Coin can first be acquired in Crustacia. A friendly Mad Monk will sell the player an Amulet of Annihilation for 10,000 Jewels, and gives them a Charm they do not already own with their purchase. If the player owns the Chocobo Egg, the Monk will give the player the Wizard's Coin; if the player does not own the Chocobo Egg, the Monk gives them that instead.
The next chance to acquire the Wizard's Coin is in the Dark Forest. In Ivor Tower South, using the well will pull up an Oglin that runs off. This Oglin can be found in the Dark Forest, thanks the player for saving them, and gives them a Charm. If the player does not own the Thug's Cloak, they receive it here; if they do own the cloak, the Oglin gives them the Wizard's Coin.
These are the only two chances to acquire the Wizard's Coin, so the player should take care they do not accidentally squander the two opportunities.