First let's establish that Rama did in fact take Indra's chariot. The incident is described in the Yuddha Kanda of the Ramayana:
Ravana for his part, standing in his chariot struck Rama with thunderbolt-like arrows, as a cloud strikes gushes of rain on a huge mountain. With an attentive mind, Rama showered golden decked arrows, looking like blazing flames of fire, on Ravana in the battle-field. The gods, celestial musicians and kinnaras thought that the battle was not equal between Rama standing on the ground and Ravana fighting from a chariot.
Hearing their nectar-like words, the illustrious Indra, the chief of celestials then called Matali and spoke as follows: "Go quickly with my chariot to Rama, who is standing on the earth. After reaching the earth, invite him to take his seat in the chariot and carry out a great service to the gods."
Hearing the words of Indra, Matali, the charioteer of Indra, offering salutation to him by bowing his head, spoke then the following words: "O Indra! Providing with green horses the excellent chariot, I will proceed then immediately and perform the duty of a charioteer to Rama.”
Then came the glorious and excellent chariot of Indra, having variegated body decked in gold, which was fitted with hundreds of tiny bells, with its pole of cat's eye gems shone like the morning sun, yoked to excellent green horses, having golden chaplets on their heads, having white whisks, covered with nets of gold, shining like the sun and bearing a flag-staff raised on a golden bamboo. Ascending the chariot, as enjoined by Indra and descending from paradise, Matali approached Rama.
Then Matali, the charioteer of Indra, along with a whip in his hand, stayed in the chariot itself and joining his palms in salutation, spoke the following words to Rama. "Indra has sent this chariot to you for your victory, O the mighty and the glorious Rama, the destroyer of adversaries! Here is the great bow belonging to Indra, an armour shining as fire, some arrows as bright as sun and a stainless auspicious spear. O heroic king! Mounting this chariot with me as a charioteer, kill Ravana the king, as earlier Indra the lord of celestials killed the demons."
Duly circumambulating that chariot (as a mark of respect) and offering his salutation, when thus spoken by Matali, Rama then ascended the chariot, causing the three worlds to shine forth with his splendour.
Now as for whether Rama violated any promise to or order from his father, I don't think he did. All Kaikeyi tells him to do, speaking on behalf of Dasaratha, is "You have to leave this coronation function and dwell in the forest of Dandaka for fourteen years, wearing braided hair and covered with a hide." And Rama responds "Let it be, as you said it. I shall fulfill the king's promise, go to the forest from here to reside there, wearing braided hair and covered with a hide." And he tells Dasaratha "I, as promised, fulfilling your command, shall live in the forest along with forest-dwellers for fourteen years." So I don't see any promise that "he shall not make use of any royal thing whatsoever".
Now you could argue wearing the "armour shining as fire" would be a violation of the "covered with a hide" part. But in any case, by the time Rama had defeated Ravana, his fourteen period of exile had ended. And it's highly unlikely that the fourteenth year ended during the battle, so presumably it was already over when Rama accepted the chariot and armour. So one way or another, Rama didn't do anything wrong.