There is a story in Mahabharata that says about some key events in the life of Vasistha. It is a conversation between a Gandharva and Arjuna.
In that conversation, it is clear that Vasistha contributed a biological son to the Ikshvaku family during Kalmashapada reign through the wife of Kalmashapada upon the request of Kalmashapada.
Recovering his power of reason, the king saluted that best of Rishis with joined palms and said, 'O illustrious one, I am the son of Sudasa and thy disciple, O best of Munis! O, tell me what is thy pleasure and what I am to do.' Vasishtha replied, saying, 'My desire hath already been accomplished. Return now to thy kingdom and rule thy subjects. And, O chief of men, never insult Brahmanas any more.' The monarch replied, 'O illustrious one, I shall never more insult superior Brahmanas. In obedience to thy command I shall always worship Brahmanas. But, O best of Brahmanas, I desire to obtain from thee that by which, O foremost of all that are conversant with the Vedas, I may be freed from the debt I owe to the race of Ikshvaku! O best of men, it behoveth thee to grant me, for the perpetuation of Ikshvaku's race, a desirable son possessing beauty and accomplishments and good behaviour.'
"The Gandharva continued, 'Thus addressed, Vasishtha, that best of Brahmanas devoted to truth replied unto that mighty bowman of a monarch, saying, 'I will give you.' After some time, O prince of men, Vasishtha, accompanied by the monarch, went to the latter's capital known all over the earth by the name of Ayodhya. The citizens in great joy came out to receive the sinless and illustrious one, like the dwellers in heaven coming out to receive their chief. The monarch, accompanied by Vasishtha, re-entered his auspicious capital after a long time. The citizens of Ayodhya beheld their king accompanied by his priest, as if he were the rising sun. The monarch who was superior to everyone in beauty filled by his splendour the whole town of Ayodhya, like the autumnal moon filling by his splendour the whole firmament. And the excellent city itself, in consequence of its streets having been watered and swept, and of the rows of banners and pendants beautifying it all around, gladdened the monarch's heart. And, O prince of Kuru's race, the city filled as it was with joyous and healthy souls, in consequence of his presence, looked gay like Amaravati with the presence of the chief of the celestials. After the royal sage had entered his capital, the queen, at the king's command, approached Vasishtha. The great Rishi, making a covenant with her, united himself with her according to the high ordinance. And after a little while, when the queen conceived, that best of Rishis, receiving the reverential salutations of the king, went back to his asylum. The queen bore the embryo in her womb for a long time. When she saw that she did not bring forth anything, she tore open her womb by a piece of stone. It was then that at the twelfth year (of the conception) was born Asmaka, that bull amongst men, that royal sage who founded (the city of) Paudanya.'"
Rama also belongs to Ikshvaku dynasty.
So, Is Vasishta an ancestor of Rama? It is not possible only if the aforementioned event happens after the Rama's departure from Earth.
Do Valmiki Ramayana or any other scripture describes Vasishta as an ancestor of Rama?