He definitely knows- he practically tells Vimes he hid the documents, with the blank expression that Vimes himself uses when lying to Vetinari.

He also knows that with his natural charisma people would follow him, but Vime's influence on him has made him think that people shouldn't just do things because Carrot tells them, and made him more distrustful of Kingship. Vimes is definitely a major factor stopping him gaining the throne, and as Vetinari himself pointed out, how long will that stop him for? (That's why Vetinari made all those allowances to Carrot- it was a bribe, not him being nice. He's too cunning to be nice)