Divine Beast Vah Ruta was piloted by Mipha of the Zora, so you should seek out the Zora people.
Go to Lanayru Tower
After completing the Seek Out Impa quest, if you go into your Adventure Log and select the Free the Divine Beasts, you will see that one of the destinations is to the east. If you want to get the map of the region where that destination is located, start by going to Ta'loh Naeg Shrine. Then go east through the forest, and you will reach a fork in the path where the right side slopes upward, and the left side goes straight ahead. Go left at that fork and continue going northeast.
You will reach another fork in the road, but don't go near the traveler. Secretly it is a member of the Yiga Clan who will try to kill you. In general, you might want to avoid travelers who seem to be trying to get your attention, because they might be from the Yiga Clan. Go north at this fork, and soon you will see some smoke in the distance. Go toward it and you will find Eryck sitting by the camp fire. You can talk to him to learn more about Zora's Domain and the Lanayru Tower.
Go north from Eryck and you will soon see the Lanayru Tower. If you go straight toward the tower, you will soon find the Rutala River blocking your way. However, you can float down to an island that sticks out from the opposite side of the river, so do that. Go northwest along this island and you will pass by a sleeping Hinox. If you wake it up, you will have to fight it (or run away). It can pull up trees and use them as weapons, so be careful.
Northeast of the Hinox, there is a wooden bridge. Go across it. As you go across, a Zora named Tona will ask you to go to the nearby Inogo Bridge, where Prince Sidon is waiting in hopes of finding a Hylian to help him.
After you are done talking to Tona, go east on the bridge. From here, you can make your way up the slope to the Lanayru Tower. You can try to climb up the north side of this hill, but it will be constantly rainy on that side. If you make your way up the grassy slopes on the south side instead of trying to climb the rocks, you will encounter Bokoblins, Lizalfos, and Moblins on your way up. If you have arrows, you can shoot at the red Xs on the wooden posts that hold up the wooden platforms that some of the monsters are standing on. They can't swim, so they will drown if you drop them in water.
At the top of the tower, you will find a Zora named Gruve, who also asks you to meet Prince Sidon on the Inogo Bridge. Gruve will show you the bridge, and you can also see a shrine near that bridge. Be sure to activate this tower (Lanayru Tower) before you leave. The shrine down below is the Soh Kofi Shrine.
Meet Prince Sidon
From Soh Kofi Shrine, go east along the road to reach Inogo Bridge. When you get there, there will be a cut scene where you meet Prince Sidon. He says that the Divine Beast Vah Ruta has been causing heavy rainfall and asks you to help. He gives you an Electro Elixir to help against the electrified enemies along the road to Zora's Domain. This starts the Reach Zora's Domain quest. After that, just follow the road as marked on the map. Sidon will talk to you at various points along the way. He will warn you of an approaching monster when you cross Luto's Crossing. Defeat the Moblin and continue along the path, and you will soon reach Zora's Domain. Sidon will ask you to go meet the king with him. Before you do, you might want to go to the lower level of Zora's Domain and open the shrine that you find there. It is the Ne'ez Yohma Shrine.
Meet the King
Back in Zora's Domain, go up the large staircase to reach the throne room. This completes the Reach Zora's Domain quest. There will be a cut scene where you learn that Vah Ruta's endless rain will make the east reservoir overflow, destroying Zora's Domain and any Hylians living nearby. The Zoras have learned that Shock Arrows are effective against Vah Ruta, but because of the Zoras' aquatic nature, they can't safely touch shock arrows, so they need a Hylian's help. The king gives you Zora Armor that Mipha herself made. However, Muzu blames the Hylians for the Calamity and for Mipha's death, so he refuses to tell you how to get shock arrows.
Go back down to the Mipha statue where you found Sidon earlier. Talk to Muzu or Sidon. After a cut scene, Link recalls a memory of Mipha. Muzu doesn't believe Link, so equip the Zora Armor that Mipha made for Link. When Muzu sees that it fits Link perfectly (which it wouldn't if it had been made to fit a Zora), Muzu finally believes you.
Get Shock Arrows
He says to go up to Ploymus Mountain, where a Lynel shoots shock arrows. You will need 20 of them. (Alternatively, you can buy them. For example, read The Hero's Sword to reach an area with a shop that sells shock arrows.) Sidon says to go up the waterfall east of Zora's Domain to get to the mountain. There is a bridge to this waterfall near the throne room. The waterfall is in Mikau Lake. Be sure that your Zora Armor is equipped, then go into that waterfall swim up it. You automatically use the paraglider after you jump at the top of the waterfall. Glide to the next section of waterfall and swim up that one as well.
You will be near Lulu Lake. Look for shock arrows in this area. Take a picture of one for your Hyrule Compendium. You might want to target shock arrows so you can find them more easily in this area, because they are scattered somewhat randomly.
If you don't find shock arrows here, go up the waterfall in Lulu Lake and then look for shock arrows in the area at the top. Then carefully go up the mountain. There will be a cut scene when you approach the clearing, and Link will hide from the Lynel.
The Lynel has very good hearing and a good sense of smell, so when you hide from the Lynel, make sure to stay far away from it at all times. It also has good eyesight, so do your best to hide behind things if the Lynel is looking toward your location. The Lynel has a very strong bow, so if he hits you with even one arrow, you might die instantly. If the Lynel sees you and starts shooting, you might want to just warp back to Zora's Domain and come back to this area again.
Carefully collect 20 shock arrows from this area. They will be stuck in trees and rocks. You will need to go up to Shatterback Point afterward. There are four shock arrows at Shatterback Point, so you only really need to get 16 shock arrows in the area with the Lynel.
Open the Divine Beast
After you have your 20 shock arrows, go to Shatterback Point and glide off of it toward the place marked on your map. When you land on that pier in the reservoir, switch to the Cryonis rune, make sure your shock arrows are equipped, and equip the Zora Armor. You might also want to change your Camera Sensitivity setting (in Options) to Very Fast.
When you are ready, talk to Sidon and he will tell you the plan.
When Sidon swims to a waterfall, press A to swim up it. As you go up, turn the camera so it is facing the divine beast. When you get in the air above the divine beast, turn the camera to look at the pink orb at the top of the waterfall. You will automatically take out the paraglider. Wait until you have descended closer to a pink orb, then shoot a shock arrow at it. You will aim in slow motion, but this takes stamina, so don't take too long.
After shooting the orb, you don't need to use your paraglider to float down. You can fall into the water and you won't get hurt. Then press A to hold onto Sidon, and immediately press L to start the Cryonis rune. Aim the camera at the divine beast so you can see the blocks of ice. Use the right control stick to aim at whichever ice block is coming toward you (using motion controls to fine-tune your aim if you have motion controls enabled) and press A to shatter the ice block before it hits you.
Once all of the ice blocks are gone, Sidon will swim to the waterfalls again. Go up whichever waterfall he approaches. He will go back and forth between the two waterfalls on whichever side he goes to, so you will have to go up one of them, even if you have already shot the orbs above both. If you have already shot the nearest orb, you can just float to another orb and shoot it, then land in the water. Sidon will come to you.
Repeat this process until all pink orbs have been shot. Vah Ruta will open its door and Sidon will take you there. You will automatically register Vah Ruta as a travel gate, so feel free to warp away if you need to heal up or prepare. Then you can warp back to Divine Beast Vah Ruta when you are ready.