It can be a bit annoying to stock up on arrows if the merchants in your game are not selling them at the moment. Luckily there are a couple of workarounds, at least for some arrow types.
Eventide Island Treasure Chest
Normally, looted treasure chests do not respawn - ever. On Eventide Island however, a whole lot of stuff just constantly reappears every time the zone is reloaded, likely to prevent the player from running out of materials during the shrine trial. I have no idea whether or not it was intended that these items continue to respawn even after the trial was completed, but if it was just an oversight then Nintendo never bothered to fix it.
There is a small isle directly east of the main island, with a mini lake in the middle. Submerged in that oversized puddle is a metal treasure chest containing 10x regular arrows, that you can fish out with the Magnesis rune. It’s exactly where the red dot is in the picture above.
After you opened the chest, save & reload on the spot. The chest will be underwater again, unopened, but you still have the 10x regular arrows you obtained. You can repeat this infinitely.
Tip: If you have the DLC, it is a bit faster to place the Travel Medallion on that mini island and just repeatedly warp to the island. This also resets the zone but doesn’t require you to save the game every time you opened the chest.
Multishot Bow exploit
This was definitely not intended by the developers, but since they never bothered to fix it, it will still work if done right. The only prequisites are that you have at least one multishot bow and at least one regular arrow to get started, as well as access to one of these places:
Stand directly in front of the lantern so that the arrow immediatly catches fire once you draw your bow. Shoot the tip of the lantern and quickly pick up the arrows again. The picture above is the approximate setup, but you don’t have to be pixel-perfect for it to work.
Equip your Multishot Bow, select regular arrows, and stand close enough to the pot that the arrow catches fire. Then shoot at the pot (around the rim) and immediatly pick up the arrows.
Repeat this until you’re satisfied, bored or your bow breaks.
Why does this work?
Multishot bows only “consume” one arrow per shot, even if they shoot 2, 3 or 5 arrows at once. Under normal circumstances, the arrows vanish after they hit something to prevent the player from easily cloning their arrows by just repeatedly shooting the ground.
Setting the arrows on fire messes up that cheat prevention just long enough for the player to quickly pick up the cloned arrows before they can disappear.
In theory, it works with every source of fire as long as the arrow catches fire shortly before you shoot it at something and you can pick up the cloned arrows fast enough, but those two locations yield predictatble results. If you just use a random campfire and shoot at a random tree, it might or might not work, even if the setup is the same.
Bomb Arrow Farming
Wet Bomb Arrows do not explode. If you find an enemy equipped with Bomb Arrows and a single-shot bow, then wait for rainfall and pester the enemy, they will start shooting Bomb Arrows at you. Just stay out of range so the arrow doesn’t hit you directly, and pick up the wet, unexploded arrows lying on the ground before you.
It is recommended to do this with the regular HUD setting instead of the Pro HUD, as the weather may switch from rainfall to sunshine while you’re distracted, and the in-game visuals are a tiny bit slower than the weather indicator next to the minimap.
One easy location is a tree camp in North Akkala, close to Skull Lake. The camp has 3 Bokoblins, all equipped with Bomb Arrows, and it rains frequently in that area. Eventually the Bomb Arrows will stop getting stuck in the ground even if the Bokoblins continue to shoot at you - in that case, simply save & reload to reset the “restriction” on farming.
Sadly I am not aware of any similar exploits for Fire, Ice or Shock Arrows. If you want to get a lot of these, you need to either repeatedly kill a lot of enemies that have elemental arrows equipped and hope that they drop some bundles as loot (for example, a lot of the Lizalfos Archers on the road through Zora’s Domain have shock arrows equipped), or repeatedly scan amiibo figurines/cards that are able to drop elemental arrows. Just buying them in shops will eventually trigger the same 50-arrow-limit restriction as for regular arrows.