You may notice that upon upgrading your shop, there is now a blue cloth and a white cloth on either side of your cash register. Customers will occasionally approach the counter without an item, standing on the right side in front of the white cloth.
When Will approaches them, there will be an option to Talk. Pressing the corresponding keybind will pause the game and open the Quest interface.
Requires Will to bring back a certain amount of a specific item. It does not tell you where the items can be found, so you may need to reference your Merchant Notebook.
Requires Will to kill 10 of a specific enemy and collect notes from them. The Notes do take up inventory space, so you'll need to manage that. It does not tell you where these enemies can be found however so you may need to do some serious dungeon diving to find them.
Turning in Quests
Will has 3 days from the day he accepted the quest to complete it (the number of days can be increased with decorations however) and, on the day it is due, Will will need to open The Moonlighter for business (having your Shop Assistant selling things does not suffice) so that the customer can collect. The customer who made the request will come in shortly after the shop opens and approach the white cloth again. Speak with them and hand over the items to complete the quest.
The items they request do not need to be in your inventory; they will collect them even if they are in a chest, but not if they are on a sales table.
Upon completion, you will receive a specified sum of gold. For fetch quests, the sum should be greater than what you would receive by selling (requires confirmation).
If you do not have the requested items or fail to open the shop on the day the quest is due, it will be marked as failed in your calendar.