[Lalwendë]

Mythology does not tend to make use of Gods/Goddesses with wings, as they often simply manifest themselves wherever needed, or take on forms which we would not traditionally associate with angels in the Christian tradition. Some Gods are manifest as animals, as trees, or even as seemingly inanimate objects. There are instances of Gods becoming winged creatures, but not becoming 'men with wings' as angels are traditionally depicted.

I think that part of the nature of 'Gods' is that they do not need to consider issues which to us would be important such as travel - as they are manifest where they need to be. There is a good instance of one of the Valar taking on a non-human form:

In reverence Yavanna is next to Varda among the Queens of the Valar. In the form of a woman she is tall, and robed in green; but at times she takes other shapes. Some there are who have seen her standing like a tree under heaven, crowned with -the Sun; and from all its branches there spilled a golden dew upon the barren earth, and it grew green with corn; but the roots of the tree were in the waters of Ulmo, and the winds of Manwe spoke in its leaves.

Tolkien writes that the Maiar do not often take on a recognisable 'human' form:

in Middle-earth the Maiar have seldom appeared in form visible to Elves and Men.

and they often take on an intangible form:

But of Olorin that tale does not speak; for though he loved the Elves, he walked among them unseen, or in form as one of them, and they did not know whence came the fair visions or the promptings of wisdom that he put into their hearts. In later days he was the friend of all the Children of Iluvatar, and took pity on their sorrows; and those who listened to him awoke from despair and put away the imaginations of darkness.

I would imagine that if one of the Valar or Maiar chose to have wings then they could have them, but there would be no need for wings. In the case of the Eagles, they took on a recognisably animal form which happened to have wings; in the case of the Istari, wings would have been an encumbrance as they hoped to pass simply as old men among Middle Earth.