This doesn't wholly replace the vastly easier solution of holding them from someone who has them, but this is more a solution that contains changes made with the Expeditions Update as well as a solution for solo players, as you may not always find a MP island and it is now doable in single player wholly with one world instance.
It should also be of note that this update does indeed remove the legendary items from being obtained from the labyrinth final room altars, so this is their location in new generated worlds now.
The update added 4 taskmasters to the game, one on each of the islands with a Labyrinth.
-The Beast Master (Badlands Island)
-The Survival Gourmand (Swamp Island)
-The The Collector (Volcano Island)
-The The Salesman (Final Island, can't think of a name cause it has all biomes, 5 difficulty)
I did not take down a list of each task for each master, so I can add them as comments or info comes in that I can find, but the long and short of it is each master has between 10-20 tasks to complete one by one. Rewards come in the form of unlocks in the Stranger's shop (trinkets and inventory slot blueprints) and Task Tokens to buy said items, but completing the full list for one of the 4 also unlocks two of the eight legendary items, with all 4 done unlocking the full list. There is enough tokens to buy one of each item from the Stranger's list with each item in the list costing one token, but be careful if the items replenish over time and don't buy something you have, take stock of which trinkets and backpack upgrades you have, or wait to complete the lists to clean the store.
As is expected each master's tasks revolve around one of the game's base mechanics.
-Beast Master is about slaying and taming animals, and also has some benchmark tasks to kill a legendary animal, the bait for whch is sold by him for doubloons.
-Survival Gourmand is all about cooking recipes which require meat, fish, fruit and foodstuff, and farming materials (emphasis on wheat and cocoa pods), and water... lots and lots of water.
-The Collector provides lists of items two at a time that you have probably sold many of over the game. Shells, bones, bugs, and some of the skeleton search items are the call here. I do remember unless the lists are random the latest ones asked for Viking Horns, Kabuki Masks, and the Sweetheart(?) Photo.
-The Salesman requires building structures such as floors, walls, etc. The good trick to this one is you can build them wherever so I just cleared a space by my main base and built in peace.
Obviously having teleporters near each master expidites this process greatly since each completed task has to be turned in by going to the master and talking to them and you can only have one active task per master at any time.
Thanks to RAS BruTaL for pointing this out: for the tame animals for Beast Master, they have to be assigned to the pen next to him for them to count on his task. Having them anywhere else will not register.