As described in this answer, Hanuman is considered to be an incarnation of Shiva. However, the Madhwa sect believes Vayu had the 3 incarnations, of which, the first was Hanuman (Sripad Madhvacharya is considered to be the third incarnation of Vayu). While many scriptures describe Hanuman to be an incarnation of Shiva, I'm yet to come across a scripture where Hanuman is seen as Vayu avatara. Instead, Hanuman is usually seen as the son of Vayu. So does any Purana describe Hanuman to be an incarnation of Vayu?
It's mentioned in this excerpt from the Garuda Purana, which lists various incarnations that Vayu took in different ages:
अतो रोचननामासौ मरुदंशः प्रकीर्तितः रामावतारे हनुमान्रामकार्यार्थसाधकः ।
स एव भीमसेनस्तु जातो भूम्यां महाबलः ॥ Garuda Purana 3.16.68 ॥
When Lord Rama incarnated on Earth, Vayu was born as Hanuman for rendering assistance to Rama.
The earliest reference to Hanuman occurs in Rig Veda which describes a divine monkey Vrisakapi in 23 verses starting 10.86. The characteristics are also common to Hanuman.
Our major reference to Hanuman is from Valmiki Ramayana (and its various translations / adaptions) and Mahabharata. Srimad Bhagavad Mahapurana and Skanda Purana also have descriptions of Hanuman and his exploits.
Certain Puranas have poisted Hanuman as an avatar of the Lord Shiva. In Valmiki's Ramayana, Hanuman was born to Anjana and father Kesari. He is also called the son of the deity Vayu, because of his role in delivering a few morsels of the sacred pudding received by Dasharatha from the Yagya Purusha of the Putra Kameshti Yagya, into the outstretched hands of the praying Anjana, due to which Hanuman was born. [I am searching for Pauranic reference to this tale and shall update in due course].
I have not come across any Pauranic reference to Hanuman as an incarnation of Vayu.