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If a Chaos Sorcerer (or any character) is equipped with one of the following:
A Warrior Familar (beginning of close combat round before blows a restruck, chosen opponent in base contact takes S5 hit)
Word Of Agony (once per game, choose a model in base contact, takes d6 s4 hits w/ no armour)
Blasphemous Amulet (at the start of friendly magic phase, enemy models in base contact with bearer (including monstrous mounts) must pass a toughness test or take a wound with no armour saves allowed)
...is he able to use the associated abilities whilst mounted in a Chariot? On that note, what about if on a monstrous mount? I never wondered until I saw that the Blasphemous Amulet specifies monstrous mounts. I suppose this could also be intended to specify that both the enemy rider and their mount are affected.

Yes, you can use these while mounted on a chariot. (I use Blasphemous amulet on a guy on a daemonic mount a lot...)

The familiar and the word randomize hits on a guy on a monstorus mount.

Yup.