There are currently three legendary dwellers you can earn normally from just playing the game and doing quests from the Overseer's office, and they are the following:

• Sarah Lyons from the 4 part quest "Climbing the Ranks"
  
• Preston Garvey From the 5 part quest "A Settler Needs Your Help"
  
• Ed the Ghoul from the 6 part quest "Searching in the Dark"

After that, the remaining 17 dwellers can only be acquired through lunchboxes.
Alternatively, the "Vault-Tec Starter Pack" you get advertised for $4,99 is guaranteed to give you one (1) legendary dweller.
Which if you'd be fine with spending money on a PC port of a phone game, would bring the number of dwellers left to get down to 16, so still long ways to go.

Big thanks to British Legends on trueachievements[www.trueachievements.com] for their guide on this, it is what i used and had as a base for this guide, with a few small additions of my own like images as well as some information they missed

• First you start by launching Fallout Shelter in windowed mode (Its easier to do it this way).
  
• Either create a new vault from scratch, make sure you have the minimum necessary power, water and food.
  OR
  
• Make sure you launch the game and your preferred vault on the first day of the daily report.
  
• After that you tab the game down and open the "Adjust Date/Time" setting in windows.
  
• Slide the "Set time automatically" setting to off
  
  
  
  
  Side note:
  Some say there's a risk of saves corrupting from time adjustment. I have not had it happen to me or met anyone who did but, keep it in mind.

• Now what you need to do is click the "change date and time" button and add 7 days to the current date.
  
• Alt tab back to the game and give it a second to reload and it should display a completed daily report.
  
• Collect all the daily rewards and the lunchbox, after that close the daily report window
  
• Alt tab back to the date/time window and enable "Set time automatically" again.
  
• Alt tab back to the game so the in-game time re-adjusts itself to the propper time and date. If you skip this step and keep adding 7 days the game risks crashing
  
• Alt tab back to the date/time window and repeat the process

This way you'll be earning infinite lunchboxes, with a rate of one every 10-20 seconds.

British Legends needed to open a total of 631 lunchboxes before they got all 20 legendary characters.

I started in a vault i had put 300 hours into, which had 14 / 20 legendary dwellers already and it took me 354 lunchboxes (3 hours of grind) before i got the last six dwellers. So even if its faster expect to grind this out for a few hours unless you're really lucky with your drops.

But, to put this into perspective:
My 354 lunchboxes would total about 6 1/2 years of daily logins to earn, assuming i didn't get any quest or task lunchboxes.

For British Legends who started on a fresh vault it would have taken over 12 years of daily logins to earn them the amount of lunchboxes needed to get all 20 legendary dwellers.

Or with the largest amount of lunchboxes you can buy in the store which is 40.
He would've had to purchase that 16 times, amounting to $319 to unlock via purchased lunchboxes.

So do not feel bad about 'cheating' on this achievement because the truth is,
..The game was rigged from the start

I made this into its own section rather than bake it in to the review, so that i can add questions as they come up and answer them to the best of my knowledge.

• Question
  After adding 7 days and waiting a short time i see the completed daily report, but I cant collect the tokens and the lunch box. I click it, but there is no lunchbox to open.
  Answer
  If the daily report shows your rewards flashing but once you click them they gray out and you don't get any reward try adjusting the time again and jump one day back in time from the date you set. This should fix the issue.

• Q: Does legendary dwellers have to all be different to count towards the achievement?
  A: No, duplicates do count. When i got my achievement i had 3 Eulogy Jones', 2 Elder Lyons and 2 Preston Garveys
  
  
• Q: Can you take a legendary dwellers' gear?
  A: Yes, if you want to put the items on some other character or scrap them to make something better, you can.
  
  
• Q: Will altering a legendary dwellers' appearance at the barber remove their legendary status?
  A: No, a legendary dweller can be altered freely and still be recognized and counted as a legendary dweller. So if you're bothered by having a bunch of clones in your vault, feel free to re-name them, change their appearance and clothing.
  

Legendary dwellers, as the name implies are rare dwellers which can either be earned in quests (of which there's only 3 you can earn.) or what you'll be getting the other 17 from, lunchboxes.
The Legendary dwellers are based off of characters from Bethesda's line of Fallout games.
In other words Fallout 3, 4 and 76.
They have predetermined SPECIAL stats, at least one of which will be 8 points or higher, and (with the exception of Amata) always come equipped with a Legendary outfit and most of the time a legendary weapon.