Is there any Adventurers League or organized play legal way to get a character with 20+ stats all around?
Note: this answer is no longer valid in season 8 rules as the Ability Score Bonuses only apply in the hardcover they exist within
There are only a few ways to increase ability scores.
Curse of Strahd
[One of the Dark Gifts] This dark gift increases the beneficiary's Charisma by 4, up to a maximum of 22.
Out of The Abyss
[The Maze Engine] On each of the engines turns, roll a d100 and consult the Maze Engine Effects table to determine what the engine does.
36-49 | Loud whispers emanate from the hollow interior of the engine. Each character in direct contact with the engine can increase one ability score of his or her choice by 2, to a maximum of 24
Available Tomes and Manuals
Tome of Understanding and Manual of Bodily Health from Curse of Strahd
Tome of Clear Thought from Tales of the Yawning Portal: Against the Giants
Tome of Leadership and Influence from Zhenatarim's Lament
Belt of Giant Strength
Fire Giant Strength available in Blood Above, Blood Below (DDEP3)
Insight available in Blood Above, Blood Below (DDEP3)
Fortitude available in Return to White Plume (DDEP06-02)
How to Actually Do it
You could try to just get lucky AND survive long enough with the method in Out of the Abyss to get your scores very high very fast (up to 13 +2 increases with perfect rolls).
The safer and more reliable option involves using your ASIs very specifically. Be very careful to avoid leaving the level ranges for each adventure as you won't be able to play those adventures once you do (try to get less XP when possible) [Advice for viable play in brackets]
order of ability scores is STR.DEX.CON.INT.WIS.CHA
- Start with Point Buy of 10.13.13.13.13.13
[applied in this way because you can't acquire an extra "+2" from the point buy alone and you will need a reasonable carrying capacity as a fighter]
- Be a Human: 22.214.171.124.14.14
[Gives +5 total stats increases (Strength ignored), better than any other race]
- You will need Fighter 19 to get 7 ASIs (applied in order to 3x Dexterity, 2x Constitution, 2x Intelligence)
[Barbarian would work too, but is less viable | I recommend starting with one level of Bard so are you are more useful in the early levels during Curse of Strahd]
- Play all of Curse of Strahd to gain those benefits (at this point you will be level 8-10): 126.96.36.199.16.18
[Play a Dexterity Fighter during this time, focus your ASIs on Dexterity]
- Play Tales of the Yawning Portal: Against The Giants (you are tackling this one a bit earlier than usual so play safe): 188.8.131.52.16.18
- Play all of Out of the Abyss (probably with a Mixed-Tier Party) 184.108.40.206.18.18 (at this point you will be level 13-15)
[You only need one of the very chance-based ability score increases with this build = 93.5% chance if you can survive long enough | apply it to Wisdom]
- Play Blood Above, Blood Below (Tier 3) for those magic items: 220.127.116.11.20.18 (at this point you will be level 15-17)
[After this, you should use Strength instead of Dexterity]
- Play Zhentarim's Lament (will need to be on the lower side of the levels to play this adventure | this adventure may not even cause you to level up if you are around level 16 but you could reach 17): 18.104.22.168.20.20
- Play any/multiple Tier 4 adventure(s) to get you to Fighter 19 (most of these are DDAL adventures found in DMsGuild): 22.214.171.124.20.20
Eldritch Knight? You could also go Eldritch Knight which is equally viable in which case you should use your ASIs for 3x Dexterity, 2x Intelligence, 1x Constitution, 1x Wisdom and use the option in Out of the Abyss on Constitution.
Magic Item count is 6, each of which is very powerful and under heavy demand so don't expect to be able to accomplish this without negotiating with other players or trading for some of the items.
A number of other builds work with using additional Ioun Stones but that makes them less reliable (and much harder to guarantee appropriate level ranges) so I chose to use this one
You can only use one Tome/Manual of any one kind on a character in Adventurer's League (DDAL FAQ v7.1 p. 8)
After rereading the FAQ for AL, there appears to be a limitation on tomes. Pg. 7 in v3. The way it is worded it is unclear as to whether tomes do not stack with themselves, or if you can only have one such tome period.
If it is the first interpretation, you could still get +2 in every stat.
So, a human barbarian would start by needing 39 stat points to have 20 in every stat. They get 18 from leveling and 12 from tomes. That means they still need 9 points to have 20 across the board. I don't think there are enough magic items to bridge that gap because of attunment restrictions.
If it is the other interpretation, you need 13 more points, and I really don't see that happening.
So, to answer the question: Not that I can see, no
The manuals and tomes described in the above question would be a legal way to get 20+ stats. Your odds of getting enough tomes to do so seem pretty low, but it would be legal.