Are there female wizards in The Lord of the Rings?

Question

I don’t remember there being any specific ones in the canon Middle-earth, but is it possible for the wizards of that universe to be female?

Answer 1

No, but....

Strictly defined, there were only five wizards in The Lord of the Rings, of whom only two appear in the books and three in the movies. These were the Istari, who included Gandalf, Saruman, Radagast, and the Blue Wizards.
All of these are male. As noted in the comments, the Blue Wizards are not explicitly called male; the Istari in general were referred to as having the bodies of Men, but Tolkien did use this word, capitalized, to refer to humans in general in various places. That said, I suspect he conceived all five as male. What this means, specifically, is that they were immortal spirits with masculine gender identity, and who had consequently taken on male forms. Speaking of a similar class of spirits, but one more powerful:

The Valar took to themselves shape and hue; and because they were drawn into the World by love of the Children of Ilúvatar, for whom they hoped, they took shape after that manner which they had beheld in the Vision of Ilúvatar, save only in majesty and splendour. Moreover their shape comes of their knowledge of the visible World, rather than of the World itself; and they need it not, save only as we use raiment, and yet we may be naked and suffer no loss of our being. Therefore the Valar may walk, if they will, unclad, and then even the Eldar cannot clearly perceive them, though they be present. But when they desire to clothe themselves the Valar take upon them forms some as of male and some as of female; for that difference of temper they had even from their beginning, and it is but bodied forth in the choice of each, not made by the choice, even as with us male and female may be shown by the raiment but is not made thereby. But the shapes wherein the Great Ones array themselves are not at all times like to the shapes of the kings and queens of the Children of Ilúvatar; for at times they may clothe themselves in their own thought, made visible in forms of majesty and dread.

_{The Silmarillion}

Unlike most Ainur, such as Sauron, who could change their form at whim, at least in the early days, and for whom it was more a sort of disguise, the Istari genuinely had proper physical bodies, with all the accompanying infirmities, albeit very long-lived.

But afterwards it was said among the Elves that they were messengers sent by the Lords of the West to contest the power of Sauron, if he should arise again, and to move Elves and Men and all living things of good will to valiant deeds. In the likeness of Men they appeared, old but vigorous, and they changed little with the years, and aged but slowly, though great cares lay on them; great wisdom they had, and many powers of mind and hand.

_{The Silmarillion}

However, there are two caveats here. The first is that there were entities very similar to the Istari, although some of different orders, that were female. Other members of the Valar and Maiar, different classes of angelic entities, weren't just male. For instance, the Valar, who are a bit like Greek gods, had seven kings and seven queens:

The Great among these spirits the Elves name the Valar, the Powers of Arda, and Men have often called them gods. The Lords of the Valar are seven; and the Valier, the Queens of the Valar, are seven also. These were their names in the Elvish tongue as it was spoken in Valinor, though they have other names in the speech of the Elves in Middle-earth, and their names among Men are manifold. The names of the Lords in due order are: Manwë, Ulmo, Aulë, Oromë, Mandos, Lórien, and Tulkas; and the names of the Queens are: Varda, Yavanna, Nienna, Estë, Vairë, Vána, and Nessa.

_{The Silmarillion}

There were also female Maiar (lesser angelic spirits) such as Melian.

Second, and more pertinent, the name "wizard", or rather its Westron equivalent, was given to the Istari by humans.

Even as the first shadows were felt in Mirkwood there appeared in the west of Middle-earth the Istari, whom Men called the Wizards.

_{The Silmarillion}

Thus, we see that (as in common English) "wizard" is simply a descriptive term, not a definition of the Istari. Only the elves (and, perhaps, a few learned humans) had an idea of who the Istari actually were. As to why they used that term, from Tolkien's own usage of the term, if we consider the term "wizardry", it seems clear that the Westron use is similar to the English one: someone who does magic. It's not limited to the Istari:

But no wizardry nor spell, neither fang nor venom, nor devil's art nor beast-strength , could overthrow Huan without forsaking his body utterly.

_{The Silmarillion}

Or:

And after Celegorm Curufin spoke, more softly but with no less power, conjuring in the minds of the Elves a vision of war and the ruin of Nargothrond. So great a fear did he set in their hearts that never after until the time of Túrin would any Elf of that realm go into open battle; but with stealth and ambush, with wizardry and venomed dart, they pursued all strangers, forgetting the bonds of kinship. Thus they fell from the valour and freedom of the Elves of old, and their land was darkened.

_{The Silmarillion}

We're even told that Sauron can do wizardry (he may be a Maia, but he isn't one of the Istari):

Then Sauron laughed; and he mocked Gorlim, and revealed to him that he had only seen a phantom devised by wizardry to entrap him; for Eilinel was dead.

_{The Silmarillion}

Pippin calls any old magic wizardry, without knowing who created it:

"Some wizardry is in it, perhaps, older and stronger than Saruman's."

_{The Two Towers}

As such, we see that the Istari were called Wizards for their magic. It wasn't just the Westron translation of Istari.

Thus, if a female Maiar, elf, or even human were known to do the sort of arts that would be known as wizardry, it's possible they would also be called a wizard (or some female equivalent; note that "witch" in Tolkien can also refer to men, such as the Witch-King).

Answer 2

There were only 5 Wizards sent from Aman to help in the fight against Sauron.

“Of this Order the number is unknown; but of those that came to the North of Middle-earth, where there was most hope (because of the remnant of the Dúnedain and of the Eldar that abode there), the chiefs were five.”

Unfinished Tales - The Istari

EDIT: Now this was a discovery for me; The 5 Istari were the leads of a bigger group sent not only to the North West, at least in the draft for an appendix to The Lord Of The Rings.

They were chosen from among the Maiar, genderless spirits, but which choose to take physical forms they find fitting to them, some male, some female, chosen by personal affinity. But it is only a "disguise", a way to experience the blessings of Arda to the full, not their actual beings.

“Therefore the Valar may walk, if they will, unclad, and then even the Eldar cannot clearly perceive them, though they be present. But when they desire to clothe themselves the Valar take upon them forms some as of male and some as of female; for that difference of temper they had even from their beginning, and it is but bodied forth in the choice of each, not made by the choice, even as with us male and female may be shown by the raiment but is not made thereby. But the shapes wherein the Great Ones array themselves”

The Silmarillion - Ainulindalë

So first answer to the question, there cannot be female Wizards; there are not even male ones, they are lesser gods with no gender independently of their chosen appearance.

The 5 Wizards are sent to Middle-earth under a set of constrains; they will not yield their full power, and they will have the appearance of old Men. Uppercase M, so they will take the shape of humans, as opposed to Elves and others.

“Among Men they were supposed (at first) by those that had dealings with them to be Men who had acquired lore and arts by long and secret study”

Unfinished Tales - The Istari

Of the 5 we know of 3, all with the shape of males. The other two, known as Blue Wizards, we know nothing except they were dressed in blue, went to the East, and never returned.

“Of the Blue little was known in the West, and they had no names save Ithryn Luin ‘the Blue Wizards’; for they passed into the East with Curunír, but they never returned”

Unfinished Tales - The Istari

So second answer to the question, one or both of the Blue Wizards might have taken the shape of a female. Nothing prevents them from doing so, as plenty of Maiar do so in Aman and there is no recorded instruction against it for their mission. We simply don't know, but the possibility is there.