I thought about making a mule in Windurst so I can do the Justice Badge quest to send it to my main but the delivery system doesn't work. Without delivery, how can we have mules? Do I need to trade with myself?

Thanks,
Bamn

You have to trade next to the character, yes.

Side note, delivery does work, it is just extremely wonky, and when you do manage to send yourself an item, that delivery slot is forever unavailable on the character used to do so. You can send items, but it often takes around 50 tries or so to be successful. Its useful at the start of the game because you can make a mule, get the racial starter ring, mail that to your main, (it will take many tries but eventually work,) then delete the mule. You can get all 3 rings on one character in this way. It would also work for regular items and quest rewards, up to a maximum of 8 items per mule. Also, before someone screams R/EX about the rings, certain R/EX items can be mailed between characters provided they are created on the same account.

TL:DR You CAN send items from a mule to your main provided you have the patience to do so and don't mind losing a delivery slot on the mule permanently. Never send anything from your main character.