This page of the FF7 Remake guide has a description of The Mysterious Moogle Merchant side mission, available in Chapter 8 - Budding Bodyguard.
This task becomes active when you rescue the children and escort them to the hidden camp. All you have to do is walk up to the boy in a Moogle Suit and trade with him. Buy a Moogle Membership Card. This will unlock the option to make purchases in the store using Moogle Amulets.
The merchant has very valuable and unique items. They aren't available elsewhere, so consider buying all/most items. Especially new weapons, accessories, Graveyard Key (required for Paying Respects mission) and items that increase AP of team members.