The Eastern and Southern Merchants are unlockable merchants in the game that will sell you rare and uncommon items you can’t easily find elsewhere. To unlock them, you have to complete the quest “Clearing the Way” on the month of the Verdant Rain Moon. This will span almost two weeks; one week for accepting the quest, and one weekend dedicated for clearing the bandits as part of this quest battle.
Once done, you should be able to buy stuff from them the following weekend. The Southern Merchant specializes in seeds, meats, and some Gifts while the Eastern Merchants will sell you baits, weapon Repair or strengthening materials, tea leaves, and gifts. Some of their items have limited stock but they do replenish their stock every month.
It’s very important to unlock them as early as you can since their items allow you to maximize the stuff you can do when Exploring the Monastery during your weekend offs. Baits allow you to fish multiple times (and earn Professor EXP in progress) while Gifts allow you to replenish your students’ motivation or increase other characters’ affinity (Support points) towards you.
Make sure to buy the items you need from them whenever you’re exploring since their inventory can’t be accessed from the Battle Preparation or weekend activity selection screens, unlike the other shops in the Marketplace.
Anna's shop is unlockable by completing the quest "The Secret Shop" on Pegasus Moon (timeskip). She sells Elixirs, Master Seals, rare accessories, shields, and stat-boosting items. Most of her items can only be bought once except for the Elixirs, Silver Shields, and Master Seals. Nevertheless, her items will help bolster your characters performance and stats.
This special merchant can be unlocked by completing the quest "Taking Care of Business" on the month of the Lone Moon (Timeskip). This merchant sells unlimited Arcane Crystals (for forging/repairing magic weapons), new gifts, rare tea leaves, and rare cooking ingredients. I've yet to confirm if this merchant is available to other routes since I encountered the quest in my Golden Deer route.