ANSWER: Regarding a gender change the Bible answer is "no" (with the possible exception of those born as hermaphrodites) but this has to be indirectly derived from Scripture. To be honest, there is no text that says, "Thou shalt not alter thy gender."
On the other hand, the Bible does make clear that God did not approve of men making themselves eunuchs (which kind of made them gender neutral). Such an act, whether carried out willingly or performed against a person's will, strips males of the ability to sire children by removing their sexual organs (Deuteronomy 23:1).
Eunuchs, even though they lost the ability to procreate, could still serve the Lord (Isaiah 56:3 - 6, Acts 8:27 - 39). Many of them, since they were slaves, had little or no choice about their sterilization in such a gruesome manner. Deuteronomy 23 seems to indicate that people should not voluntarily choose to mutilate their reproductive organs.
A fundamental limitation of modern procedures for changing genders is that the organs implanted or created are not fully functional. A man who tries to become a woman will not be able to give birth, nor will a woman who becomes a man be able to sire children through intercourse. They are merely trophies or symbols, not a functional reality.
Therefore, someone who feels he is a woman trapped in a man's body, or vice versa, needs to visit someone specialized in psychological counseling before he or she would surgically mutilate himself or herself in order to "change" their gender.
It should also be noted that gender is not just a matter of the reproductive organs. Its imprint is on every cell through the XX or XY chromosome, including in the brain. As a result, although someone can take hormonal drugs and undergo surgery, he or she cannot really change his or her gender at a fundamental level. The mindset imprinted in a developing fetus' brain by the presence of testosterone, for example, is not removable decades later long after birth by taking estrogen.
It is much better for people to learn to accept who and what they are physically rather than trying to change their gender surgically, since the results are so crudely limited compared to what would theoretically have to be removed, altered, or both in order to transform someone's gender.
The differences we see between men and women are not mainly created by society but reflect biologically driven (which was created by God) realities. By accepting the teaching of Scripture, we merely accept what we could discover and reason from nature based upon anthropological and sociological studies.
In closing, the Bible generally does not approve of a gender change. Although we cannot cite a direct text on the matter, we can make a conclusion based on the evidence available.