From what I read and understood on the forums, quests will not require you to commit illegal acts to progress through.
However, there is a quest in Koeglin Village in Stormhaven that requires you to investigate the Constable's house. That house is locked and it is of course a crime to pick the lock. Even though lockpicking is simple enough, I wonder whether this was intended.
Thank you.
Patroller and Editor at
UESP