In this case, you can say "I promise to find it for you." or "I promise you, I'll find it." This is because the normal indicative verb "promise" does not take the preposition to. So you do not "promise to someone", you "promise someone".

When talking directly to the person, you might leave our the first "you" in this example. "I promise to find it for you". Or the second: "I promise you I'll find it."

'I promise to find it for you' sounds better. The phrase 'I promise you' alone is usually used for emphasizing that what you are saying is true. So in general you 'promise to do sth' or you 'promise sth to someone / promise someone sth'. You can also 'promise someone that...'

Yes! I promise you that I will find it for you. I promise you that I will take care of you no matter what happens.