What does "take a hit" and "in the order" mean?

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Take a hit = be reduced. It's slang.
To understand this, think about HP (health points) in RPGs. When you take a punch (a hit), your health is reduced.
So, here, "the contributions that the abbey receives would take a hit", means "the contributions that the abbey receives would be reduced".

In the context of this sentence, an "order" is a group or cult. Here, I suppose, we're talking about the order of monks (group of monks).
If Angelo wasn't in the group (the order), then the abbey would receive less contributions from wealthy people (contributions from wealthy people would take a hit).