You can upgrade Mr. Kit's store by purchasing a Store Upgrade Catalog from him.

Purchasing a Store Upgrade Catalog will close Mr. Kit's store for the rest of the day, but rarer items will become available to purchase when it reopens the next day. This will not prevent you from selling items from your inventory though.

Mr. Kit's store can be upgraded multiple times, but the price will increase with every upgrade. Note that you may also need to progress to a certain point in the story before you are able to purchase the Store Upgrade Catalog.