I'm looking for a word to describe someone who is very driven by morality, constantly trying to do what they believe is morally right.
This word would be used in a sentence like "I really admire her; she truly is a (...) person."
Roget's Thesaurus offers the following:
Those words may have slightly different connotations, but they all essentially mean that the person is trustworthy because of their moral fiber.
Someone who is zealous about something has a lot of passion for it. Someone who is morally zealous would be someone who is all about sticking to their moral principles:
I really admire her. She truly is a morally zealous person.
I assume from your example you are looking for a positive word. As others have already provided examples of this, I won't add to them. However there is an expression with a usually negative connotation that fits and hasn't been suggested:
To paraphrase Adam Ant: They don't drink, they don't smoke - what do they do?
I really admire her, she truly is a good person.
2a (1) : virtuous, right, commendable a good person
If you want to lean towards a negative connotation (which, admittedly, does not fit your example sentence):
do-gooder: A well-meaning but unrealistic or interfering philanthropist or reformer.
In German there's also the (rather condescending) word Gutmensch (literally: good human) which I thought was in use in English as well, but neither the usual dictionaries nor google ngram know about it..