Arrows are extremely important to have on hand in Tears of the Kingdom. Without them, certain monsters will be extremely difficult to defeat. Fortunately, the "Fish for Fletching" side quest in Rito Village is a great way to farm arrows for those that may be running low.
"Fish for Fletching" is a fetch quest, but all the resources are nearby if players know where to look. Like some Tears of the Kingdom side quests, this will only unlock if players have completed the Wind Temple. Until then, it may be best to hold on to any 
Brightcap, 
Chillfin Trout, and 
Glowing Cave Fish.
Where is Rito Village?
Players can find Rito Village near coordinates -3596, 1801, 0209 On a first visit, the bridge leading to the village will be broken. Link can either build his own bridge (with Ultrahand) by cutting down the nearby trees, or he can glide in (with the paraglider) if he's up high enough. Since "Fish for Fletching" won't be ready until Link completes the "Tulin of Rito Village" main quest, it may be some time before newcomers can use this method of gathering arrows.
Once players defeat Colgera and Tulin awakens as the Wind Sage, return to Rito Village and go up to coordinates -3655, 1763, 0214. There will be a pink/red female Rito named Bedoli standing by a group of crates. Speak to her and hear her out to accept the "Fish for Fletching" side quest.
Fish for Fletching Walkthrough
In order to complete "Fish for Fletching," Bedoli will need three Glowing Cave Fish. As their name indicates, these fish won't be found in regular bodies of water. Instead, look inside caves or wells. They glow a bright white/purple, so they won't be hard to find.
Players can check the Snowfield Stable Well (-1630, 2587, 0234) or they can go to the West Lake Totori Cave (-4005, 2031, 0191). The West Lake Totori Cave is where players will need to go for "The White Bird's Guidance" Shrine Quest:
From Vah Medoh's Perch in Rito Village, glide west towards what looks to be a white bird in the mountainside. The "bird" is actually just how the snow fell on the mountainside, and it's "looking" directly at the West Totori Cave entrance.
Head inside, and there will be a cavern lake with Glowing Cave Fish and Chillfin Trout. There are also Brightcaps along the walls and a Bubbulfrog deeper in the cave. The West Totori Cave is closer to Rito Village and it also has all three items that Bedoli will need.
When fishing, players should use arrows. Do not fuse 
Bomb Flower to arrows as this will turn the fish into Roasted Fish/Trout, which no longer count as Glowing Cave Fish/Chillfin Trout. Either shoot them with regular arrows, or dive in the waters and swim after them.
Fish for Fletching Reward
Give Bedoli the three Glowing Cave Fish to get x10 Arrows. Now that this side quest is complete, players can speak to her numerous amount of times to get x10 arrows as long as they have the requested materials. Bedoli will cycle through Brightcaps, Glowing Cave Fish, and Chillfin Trout.
West Totori Cave has all three materials, but players can also find the Chillfin Trout at Zora's Domain (3299, 0424, 0112) or Tama Pond (-2997, 1895, 0307). Due to the rarity of Chillfin Trout, it may be best to hold onto them and use them for upgrading the Frostbite Armor or for cooking heat-resistant meals. Meanwhile, Brightcaps and Glowing Cave Fish are extremely common in just about any cave or well.
Arrows are fairly common in Tears of the Kingdom as bundles of them can be retrieved from breaking crates or barrels. They can also be bought in bundles of five at the general stores. Players will want to trade materials to Bedoli as long as they don't need them to upgrade armor. Regardless, this is a very effective and free way of farming arrows if players desperately need them.
The Legend of Zelda: Tears of the Kingdom is available for Nintedo Switch.