I believe Vyasa appears in Mahabharata. Is there anything like that in Ramayana?
Yes Valmiki appears in the Uttara Kanda, where he provides shelter in his ashrama to the banished Sita. Later events take place at his ashrama with Valmiki playing an important role in the bringing up of Lava and Kusha.
Refer link : https://en.wikipedia.org/wiki/Ramayana#Uttara_Kanda
Yes, not only is Vālmīki a character in his Rāmāyaṇa he even meets Rāma in person and gives testimony of Sītā's unblemished character.
Chapter 7 (Uttara Kanda) - Sarga 87
When night was over, the king, Raghava, went to the sacrificial ground and summoned all the immensely energetic rishis. Vasistha, Vamadeva, Jabali, Kashyapa, Vishvamitra, Dirghatapa, the great ascetic Durvasa, Agastya, Bhargava, Shakti, Vamana, Markandeya, the great ascetic Moudgalya, Bhargava Chyavana, Shatananda, who knew about dharma, the energetic Bharadvaja, Suprabha, Agni's son, many other sages who were firm in their vows and all the kings who were tigers among men assembled. Filled with curiosity, the extremely valiant rakshasas and the immensely strong apes all these great-souled ones also assembled. Kshatriyas, vaishyas and thousands of shudras all of them also assembled, to see Sita take the pledge. All of them came there and were like mountains made out of stone. They heard the sage coming quickly, accompanied by Sita. With her face cast downwards, Sita followed the rishi at the rear. Her voice choked with tears and in her mind, she joined her hands in salutation before Rama. The beautiful one could be seen to advance, following the brahmana. Sita was behind Valmiki and loud roars of praise arose. There were sounds of uproar everywhere. On seeing her immersed in this great misery, they were overcome by grief. Some praised Rama. Others praised Sita. There were yet others who praised both of them. The bull among sages entered in the midst of that crowd of people, with Sita as his companion.
Valmiki spoke to Raghava. 'O Dasharatha's son! This is Sita. She is excellent in her vows and follows dharma. Without having committed a crime, she was abandoned near my hermitage. O Rama! O one who is great in vows! That was because you were scared about people censuring you. You should grant your permission, so that Sita can establish your trust in her. These twins were born as Janakee's sons. These invincible ones are your sons. I am telling you this truthfully. O descendant of the Raghava lineage! I am the tenth son of Prachetas. I do not remember having uttered a falsehood. These are your sons. I have performed austerities for many thousands of years. If Maithilee isn't innocent, I will not reap the fruits of those. O Raghava! Had I not known Sita to be pure through my five senses and through my mind as the sixth, I would not have accepted her near that waterfall in the mountain. She is pure in conduct. She is devoid of sin. Her husband is her divinity. However, you are scared because people condemned you. She will instill confidence in you.'
Valmiki appears in Bala Kanda and Uttara Kanda.
The Ramayana starts with the conversation between Sage Narada and Valmiki. Valmiki inquires whether there is any man in the world with qualities of a God. He wanted to know sixteen qualities.
A thoughtful-meditator, an eternally studious sage in scriptures about the Truth and Untruth, a sagacious thinker, and a sublime enunciator among all expert enunciators is Narada, and with such a Divine Sage Narada, the Sage-Poet Valmiki is inquisitively enquiring about a man who is a composite for all merited endowments in his form and calibre. [1-1-1]
He inquires whether there is a man with godly qualities in the world and states 16 qualities he is looking for. Narada maharshi names Sri Rama Chandra and narrates the story in brief. Valmiki gets curious and he gets a divine message from Lord Brahma to compile the epic. He bestows a boon that all the events happened in the past, present incidents of the characters in the Ramayana whether revealed or unrevealed just by thought. In this way, Valmiki know what happened exactly.
He appears in the Uttara Kanda of Ramayana in several places. He gives shelter to Sita Devi who was in exile. Valmiki maharshi took care of her and her children too. He named him Lava and Kusha.
There upon taking Kusa (the upperpart of the grass) and Lava (lower part) the great ascetic Valmiki made those arrangements. And giving Kusa sanctified by those Mantras unto the hands of elderly woman he said, "do ye rub the person of the elder boy with these"; and giving them Lava he said “do ye rub the person of the younger brother with this. And according to this I shall name the first Son Kusa and the second Lava
More information on the naming ritual of Lava and Kusha at Who was elder amongst Lava and Kusha ( twins of Rama and Sita?.