Minecraft Dungeons

So as the other solution is a bit bland and doesn't give you all the options to get as many emeralds you can during one playthrough ill tell you all ways you can acquire emeralds and it doesn't mean smashing vases for 10 hours straight!
( Other Solution Was Deleted )
If you play this game rather often this will just unlock through natural progression of the game and you will unlock this through a few playthroughs but to get this as soon as you can follow these tips, good luck!

1. Try to get gear with the Prospector Enchantment and upgrade it to max level, fully upgraded gives you a 300% extra chance to get Emeralds for killing an enemy and you could combine multiple gear items with Prospector for even more. (credit to bgiesing for ^^^)
2. you can salvage Artifacts, Armor, Ranged and Melee Weapons for emeralds (note while salvaging have a double check for 3 key things you need for a achievement called)

3. In your camp have a look around and there are chests with 50 Emeralds in them so keep your eyes open when around your camp!
4. in your missions there is a pig that sometimes spawns maybe 2 to 3 times a mission with a 'Fancy Chest' on top of him and that contains emeralds so shoot him with a bow and the emeralds will drop, i do recommend killing the pig for more loot... poor piggy :(
5. Mobs will sometimes drop emeralds on killing them around 3 at a time so not very worth it but for killing Evokers and Endermen it might be more. (the drop of emeralds vary!)
6. To make sure you get all opportunities to collect all emeralds on one mission i suggest going to every part of the map even if its away from the objective you can also find hidden chests that will sometimes give you emeralds it also can bank you another achievement
7. there are vases scattered around that drop emeralds but not many so if you see them feel free to smash them but don't make it essential to smash all of them, only if you come across them!

if i did miss something in this solution you thought could be helpful feel free to comment and tell me. id appreciate your feedback than a dislike.