What are the conditions to get the “Who Needs Tygan?” achievement?

Apparently, coming from someone who did check in the game’s files for the achievement conditions :

Of note is that this only applies to the final mission, you can
happily upgrade throughout the game, just make sure your squad going
into the final mission is downgraded. (Tip: Psionics/Dominate is
AWESOME).

Only Tier 0 items:

• Assault Rifle, Pistol, Shotgun, Cannon, Sniper Rifle, Sword, Gremlin Mark I, Psi Amp, Grenade Launcher
• Nanofiber Vest
• Medikit
• Frag Grenade
• Flashbang
• Smoke Grenade
• PCS Upgrades
• Weapon Mods

Exceptionally, the following upgrades are allowed: (since you can’t
downgrade)

• Alien/Plasma Grenade (Permanent upgrade to Frag Grenade)
• Nano Medikit (Permanent upgrade to Medikit)
• Smoke Bomb (Permanent upgrade to Smoke Grenade)

Forbidden Items include (not comprehensive – ANYTHING not in the list
above may be bad):

• Skulljack
• Battle Scanner, Mindshield, Mimic Beacon
• ANY Special Ammo (Talon, Venom, Dragon, AP, etc.)
• ANY Special Armor (EXO Suit, Warden armor etc.)
• ANY Special Grenade (Gas, EMP, Acid Grenade/Bomb, etc.)
• ANY better weapon (Advanced Grenade Launcher, Arc Blade, etc.)

In essence, the game checks the inventories of all soldiers and makes
sure that you only have Tier 0 items, with the exception of the three
permanent upgrades (Alien Grenade, NanoMedikit, Smoke Bomb) and any
drops you got throughout the mission.

Interestingly, the Nanofiber Vest is allowed as well.

Source : XCOM Reddit