For example, if an Elder Cathar it exiled by one of Helvault’s abilities, the triggered ability of the Elder Cathar will not trigger. Similarly, if a creature is exiled by Helvault, the ability of Galvanic Juggernaut will not trigger.
Helvault does not trigger ‘goes to the graveyard’ or ‘dies’ triggers.
Helvault has two activated abilities, both of which exile creatures. Exiling means to put a card into the exile zone. Creature that are exiled do not go to the graveyard and will not trigger relevant abilities that trigger when a creature dies (because ‘dies’ means ‘is put into a graveyard from the battlefield,’ and ‘exile’ is not a graveyard).