Whenever Lord Vishnu descended on earth in human form, all demigods like Indra,Vayu, Surya, Adi Shesha and others also came along with Him. When Lord Vishnu took Ramavatara, Adi Shesha was born as His younger brother Lakshmana, Vayu was born as Hanuman, Surya was Sugriva etc. When Lord Vishnu took Krishnavatara, Adi Shesha was His elder brother Balarama, Indra was Arjuna, Vayu was Bheema etc. Even Goddess Lakshmi was born as Seeta and Rukmini respectively during Ramavatara and Krishnavatara. Why is it that Garuda, the vahana of Lord Vishnu, never took birth to assist Him during His avataras?
Garuda is the foremost vahanas of Lord Vishnu. He is the most favorite one too.
Though other devatas, nitya suris like Adi Shesha, Shankha, chakra incarnated as different characters during Ramayana and Mahabharata times, Garuda didn't take a direct avatar. However, he was always ready to help Lord Vishnu whenever needed. He descended to earth (literal meaning of an avatara) when needed. It is evident from various incidents.
Churning of Milky Ocean
Devas and danavas agreed to churn the milky ocean and chose Vasuki as rope, Mount Mandhara as churning rod. Lord who took birth as Ajita lifted mount mandhara with one hand on the back of Garuda. Garuda took that mountain near the water (? where did water come from?).
avaropya giriṁ skandhāt
suparṇaḥ patatāṁ varaḥ
yayau jalānta utsṛjya
hariṇā sa visarjitaḥ Bhagavatam [8.6.39]
Thereafter, Garuḍa, the chief of birds, unloaded Mandara Mountain from his shoulder and brought it near the water. Then he was asked by the Lord to leave that place, and he left. [8.6.39]
Nagapashavimochana from Ramayana
Rama and Lakshmana were severely wounded by the arrows of Indrajit. Since he couldn't defeat the two princes in his manifest form, he sought to his magical powers and wounded the Rama, Lakshmana and monkey army with his magical arrows. He shot venomous snakes in the form of arrows. Due to this, Rama and Lakshmana fell unconscious on the ground. Sushena, the vanara who was conversant in treatment asked Sugreeva to bring herbs present in the milky ocean which could revive Rama and Lakshmana. Then a great wind arose. Then appeared Garuda, king of birds. Just by his sight, the snakes which bind the bodies of Rama and Lakshmana went away into the ocean quickly. Garuda appeared there. By his touch, the wounds of two princes were healed. Their energy, valour, memory was restored.
In the meantime, a great wind arose, accompanied by massed clouds and lightning, whipping up the salty waves in the ocean, causing the mountains to tremble as from an earthquake. Large trees on the sand-banks had their branches broken by the mighty stroke of Vata's wings and fell headlong into the briny waters of the ocean. The snakes became frightened. The snakes inhabiting there and all the marine animals plunged quickly into the briny ocean. On beholding Garuda the eagle which came there, the serpants who bound those two warriors Rama and Lakshmana in the form of mighty arrows, fled away..... Their wounds, touched by Garuda the eagle, were healed. Their bodies soon became smooth and well-rounded..... O, Rama! I am your dearest friend Garuda dear as your own breath moving outside I came here for the purpose of helping you, both......Hearing about this incident, I have hastened all at once, duly cherishing the love and friendship (for you)....... O, hero! You need not be so curious to know the casue of my friendship. You will know of it, once you have accomplished success in battle. [Yuddha Kanda Sarga 50]
During Krishna avatar
There are several incidents of association between Krishna and Garuda. Garuda Purana is a conversation between Krishna and Garuda. He became mount to Lord Krishna while slaying rakshasas like Narakasura etc., Garuda also accompanied Krishna in thee incident of Parijatapaharanam. There could be many other incidents too.