As title implies, I am looking for Champions that have a Zealot-style background to them.
They fight for what is right and will let nothing get in their way!

Would any of you be so kind as to help me? [I am still looking through the lore myself so that I'm not lazy]

Vayne taken up to eleven

Vayne doesn't really follow this. She looks for evil and shoots it, that's not really the same. I just took a look at the champion list, and I don't really think there is any champion like that. It seems like what you're looking for is a Paladin-Type character, but LoL doesn't really have that because a paladin doesn't really make for great storyline, since it's really cliche and simple. Most champions either fight for country(Like J4 or Darius), for the pleasure of it(Jax or Evelynn), or for some ideals or objectives they have(Like Rengar or Yorick).

Well since "right" is subjective, Darius is pretty single-minded about his desire to push the Noxian way of life.

Almost anyone from Ionia too.

Sounds like Riven? A lot of people would fit that. Lee was willing to burn himself for his principles, and Shen went through that process of his to become the Eye of Twilight.

What about, like, Kayle?

Syndra; she is uncompromising, powerful, confident, and confident in her own decisions. Considering Zealot does not necessarily refer to a religion or cult, Syndra's own ideals make her a zealot.

Nasus; his personality and his quotes are dripping with his dedication, resolve (to stop his brother), and toward the balance of life and death.

Kassadin is, without a doubt, the best example of a zealot. He wishes to protect the world from the Void at all costs using the powers he gained it. Malzahar is a zealot on the opposite end of the spectrum which is clear from the fact that he has his own cult which worships the Void.

I realize "fanatic" does not necessarily mean the same thing as "zealot", but they are definitely similar.

I think the only 'zealots', with its religious connotation, in the League would be Diana or Malzahar.

Zealot is generally a negative term, so fighting for what is 'right' is probably not what you'll get from it. They think they're right, sure, but they're also unyielding, which is a bad thing. Diana thinks she's right, and kills people who think otherwise. Malzahar thinks he's right, and is calling for the world to end because of it.

The ones who are fighting for justice and balance and righteousness in the lore wouldn't be considered zealots. The majority of them are actually pretty logical and rational. Zealots would not be.

Diana.
Though "right" is kinda dependent on what you think about the Rakkor.