Is there any mention of Kula Devi or Devata name of Shree Rama in Srimad Valmiki Ramayana?
Apart from the other answer already given, it is also mentioned that Devi Kausalya, worshipped Lord Vishnu for the welfare of her son Sri Rama.
Ayodhya Kanda, sarga 17 (critical edition)
kausalyApi tadA devI rAtriM sthitvA samAhitA | prabhAte tv akarot pUjAM viSNoH putrahitaiSiNI || 6
Queen Kousalya had spent the night in a self-controlled way. In the morning, for the welfare of her son, she was worshipping Vishnu.
It can be assumed that Lord Vishnu is the kula devata.
It's Lord Vishnu/Narayana. Rama worshipped Narayana before his coronation ceremonies in Ayodhya Khanda -
After Vasistha left, Rama took bath and meditated on Lord Narayana with undistracted mind along with his wide-eyed wife, Seetha. Taking the vessel with clarified butter on his head as per scriptures, he offered to Lord Vishnu the clarified butter, by dropping it into the blazing fire. Rama ate the remainder of clarified butter after finishing the sacrifice, which he performed for his own good, silently meditated on Lord Narayana with controlled mind and slept along with Seetha on a properly laid bed of Kusa grass in a splendid temple of Lord Vishnu.
Ayodhya Kanda, sarga 6
Kaushalya was also worshipping lord Vishnu -
At that time, Kausalya having spent the whole night with steadfastness, who performing worship to Vishnu, at dawn, for the welfare of her son.
Ayodhya Kanda, sarga 20
Bhagwaan Jagganatha ji, who is Shri Hari Vishnu ji himself, is the kula devata of Raghu vansha clan.
This is further strengthened on the fact that we have various references to Siya-Ramji and their mother kaushalya devi worshipping Narayan or Vishnu Bhagawaan, as we can see from other answers on this question too. Gita Press Ramayan interprets Jagganatha word as Vishnu ji. We can read it through this link too - https://archive.org/details/fdag_shrimad-valmiki-ramayan-with-hindi-trans.-part-2-series-no.-76-gita-press/page/n851/mode/2up?view=theater
Bhagawaan Shri Ram ji himself answers this in the Ramayan by Maharishi Valmiki in Book 7 called Uttara Kanda.
बिभीषणमथोवाच राक्षसेन्द्रं महायशाः । यावत्प्रजा धरिष्यन्ति तावत्त्वं वै बिभीषण । राक्षसेन्द्र महावीर्य लङ्कास्थस्त्वं धरिष्यसि ।। ७.१०८.२७ ।।
यावच्चन्द्रश्च सूर्यश्च यावत्तिष्ठति मेदिनी । यावच्च मत्कथा लोके तावद्राज्यं तवास्त्विह ।। ७.१०८.२८ ।।
शासितस्त्वं सखित्वेन कार्यं ते मम शासनम् । प्रजाः संरक्ष धर्मेण नोत्तरं वक्तुमर्हसि ।। ७.१०८.२९ ।।
किञ्चान्यद्वक्तुमिच्छामि राक्षसेन्द्र महामते ।। ७.१०८.३० ।।
आराधय जगन्नाथमिक्ष्वाकुकुलदैवतम् । आराधनीयमनिशं सर्वैर्दैवैः सवासवैः ।। ७.१०८.३१ ।। तथेति प्रतिजग्राह रामवाक्यं विभीषणः । राजा राक्षसमुख्यानां राघवाज्ञामनुस्मरन् ।। ७.१०८.३२ ।।
Thereafter he said to Bibishana, the King of the Rakshasas:—
“As long as people exist, O Mighty Lord of the Rakshasas, do you remain in Lanka. As long as the moon and sun continue to shine on the earth, as long as they speak of me in the world, so long shall thine empire endure; this is the will of your friend, do you obey! Govern your people according to justice and do not question this further. One thing more I wish to say to you, O Mighty Monarch of the Rakshasas, do you worship the Divinity of the House of Ikshvaku, the Guide of the Universe, Shri Jagannath, whom the Devas themselves with their leaders perpetually adore.”
“So shall it be,” answered Bibishana,
Ikshvaku being the first king of Shri Ram ji's dynasty, and hence it is called Ikshvaku-vansha.
And thus as told by Bhagawaan Ram ji, the kula devta of their clan is a specific form of Vishnu - Lord Jagannath (which is Vishnu himself, if I'm not misplaced).
I have earlier concluded that the word Jagganatha as Puri exclusively, and I understand that it might be inconclusive for some of us. But my conclusion were based on what my guruji lectures, he gave cyclic nature of time and other reasons like Brahma ji praising Ram ji as Krishna (an avatara which is yet to come) after the Lanka war.
Anyway, the major question was who is the kula devata of Ramji and the answer is it is Bhagawaan Jagganatha ji, who is Shri Hari Vishnu Bhagwaan ji himself. Even if we do not specifically conclude the Jagganatha word as the god of Puri temple, it is Vishnu ji only (as further supported by Gita press hindi Ramayan translation).