This page explains how to start and complete the Special Delivery side quest.
How to Start the Quest
First, you need to complete the Divine Beast Vah Ruta Dungeon. Then you can talk to Finley at the Bank of Wishes in Zora's River. The Bank of Wishes is about halfway between Boné Pond and Zora's Domain, on the west side of the river.
How to Complete the Quest
Follow the letter along the river. It will be stopped by a wooden fence that you can break with a weapon (but be sure not to hit the letter container itself).
After the letter falls down a couple of waterfalls, be sure to kill the Octorok along the river. If the letter gets stuck in a dead-end, pick it up and move it to the next part of the river, but keep killing any Octoroks along the way. There will also be a couple of Lizalfos that you may have to deal with. Then it will eventually reach the monster camp near Boné Pond, where there is another Octorok. Underneath the monster camp bridge, you will need to destroy the wooden barrier to let the letter continue on its way. There are more monsters after this barrier.
In the end, the letter arrives at a camp. Approach the letter after it reaches the beach.
After Sasan leaves, go back to Zora's Domain and go to the quest destination. Talk to Finley to complete the quest.
Quest Reward
You get a Gold Rupee for completing the quest.