From my experience, the Morality items chest in Voic Crusades only has a chance of giving you one; it can totally give you none a few times in a row.This is really annoying if you also have a high item modifier from doing all the crusade missions, less so if you just did a minimum number of missions.

It's not guaranteed. It Can give it and it gives at high loot chances. With 1000+% increased loot (full clear+bonus missions + fortune buff + tarot cards) I had an average of 3 morality items per run. (With 1 as lowest and 7 as highest). With only a partial clear at bonus of 250-500% it's possible to get no morality items.

As far as I understand, tarot cards do not influence the final chest drops, only drops in the missions.

As per the compendium: "The bravest of Inquisitors can apply Tarot Cards to these missions, further increasing the rewards gained from individual missions (Tarot Cards do not modify the stacking loot bonus on the Void Crusade map)."

In fact, if you apply tarot cards to the supreme mission, their effect will end once the mission is done and before you teleport to the loot room (you get the mission complete pop up and summary before you teleport). Unless there has been an update that is not reflected in the compendium...?