In order to unlock all recipes, it's necessary to complete all 20 Photo Missions and deliver at least 58 different kind of materials. Farming materials is considerably easier in Kingdom Hearts III. Not only is the drop rate generous, it's also possible to farm almost all required materials from Battlegates (post-game challenges). In case you want to speed this up even more, use the Favourite Deputy keyblade and equip a Lucky Ring, which is a random reward for posting Prize Postcards at the red mailbox near the Moogle Shop in Twilight Town. Prize Postcards are given by the Moogle after making purchases at the shop. At first it's possible to obtain around 10-12 of them just by purchasing 1-2 ingredients and exiting the store.

Objects you have synthesised already are listed in the Gummiphone's Synthesis section. If you have patch 1.04 installed, a check mark will also appear next to items in the Moogle Workshop that have already been synthesised.

One thing the Synthesis Guide doesn't mention is where to farm Hungry Gems. Hungry Gems and Frost Gem are obtained in Arendelle as a possible drop from Frost Serpents. For a chance to obtain Hungry Gem you must destroy their tails. For a chance to obtain Frost Gem you must destroy their wings. Another possible location for Frost Gems are the snowballs scattered across the map in Arendelle, you must jump on them and roll into a wall to destroy those.

Below is a video that shows how to farm Frost Serpents.

Farming Frost Gems and Hungry Gems