Armored Core - For Answer

Salineos and 051ANNR:
To unlock the Salineos missiles and 051ANNR rifle you have to complete the
"Defeat the White Glint" mission at the end of Chapter 2.

EC-0307AB:
To unlock EC-0307AB triple back laser cannons, you have to complete
the "Attack on Carpals" mission in Chapter 3.

All purchasable parts:
To unlock all purchasable parts you have to complete both the Collared and
ORCA story modes.

07-Moonlight (laser blade), ASB-0710 (additional boosters shoulder piece):
To unlock 07-Moonlight laser blade and the ASB-0710 boosters shoulder piece you
have to defeat Orca Rank 5 Next Shinkai.

White Glint Head, Core, Arms, Legs, and Head (top) Stabilizer:
You have to complete the "Defend Line Ark" mission at the end of Chapter 2
(White Glint does not have to survive).