Why do you say there's "no evidence"? How are you defining evidence, and would you maintain the same skepticism if it were commonly held that he was straight but there were no wives, no illegitimate children, no known mistresses?

For hundreds of years, people did not really talk about Leonardo da Vinci's private life. It was generally accepted that he was a very great man - one of THE great men of "Western civilization" - and of course it would be lesé majesté to discuss the potential of homosexual behavior on his part. People did not actually talk about homosexuality as an identity; they saw men having sex with men (and women with women) as immoral behavior that indicated character deficiencies. In the second half of the nineteenth century, however, same-sex attraction began to be studied as a medical/psychological issue, and by the turn of the century researchers were able to treat it as a way of being without a moral component. This is the atmosphere in which the Florentine records showing that Leonardo da Vinci was formally accused of sodomy were finally published in 1896.

In 1910, Sigmund Freud published a small book titled Eine Kindheitserinnerung des Leonardo da Vinci - "A Childhood Memory of Leonardo da Vinci". (This is a few years after Elisar von Kupffer wrote a book on one of Leonardo's students, Antonio Bazzi, known as Il Sodoma because of his proclivities; Freud was certainly aware of it.) This book begins by noting that while people have talked quite a bit about Leonardo's work, there has been very little attention given to his personal life and personality, even in writings by others from his lifetime. There's a lot of Freudian analysis in it that I am not remotely capable of unpacking, but basically: Freud saw a struggle within Leonardo, shown by his tendency not to finish his paintings; he explored a particular story Leonardo recounted later in life, of having a bird of prey put its tail in his mouth as a baby by relating it to the desire to perform fellatio and the act of nursing; and he explained that sons of absent fathers (like Leonardo) often ended up being attracted to men instead of women. Outside of this kind of shaky interpretation, Freud pointed to the known circumstances of the artist's life:

It is doubtful whether Leonardo ever embraced a woman in love, nor is it known that he ever entertained an intimate spiritual relation with a woman as in the case of Michelangelo and Vittoria Colonna. While he still lived as an apprentice in the house of his master Verrocchio, he with other young men were accused of forbidden homosexual relations which ended in his acquittal. It seems that he came into this suspicion because he employed as a model a boy of evil repute. When he was a master, he surrounded himself with handsome boys and youths whom he took as pupils. The last of these pupils, Francesco Melzi, accompanied him to France, remained with him until his death, and was named by him as his heir.

This was taken quite seriously. Hirschfeld (the founder of the Institut für Sexualwissenschaft) published a summary of the book, and Ellis listed Leonardo along with Muret, Erasmus, Michelangelo, and a number of other historical figures as homosexual in the "sexual inversion" volume of his Studies in the Psychology of Sex. And really, it has been historical consensus since this point.

Florence was known in its time for being more accepting of romance and sex between two men (or rather, for their being practiced widely despite official condemnation), to such an extent that it had a reputation throughout Europe. This was discussed blatantly, with only minimal cover, by chroniclers, visitors, and locals, usually negatively by the first two and both positively and negatively by the latter. This has been known for a long time, but owing to the attitudes in my first paragraph, it was taboo for historians to entertain it seriously for centuries. Michael Rocke states in Forbidden Friendships: Homosexuality and Male Culture in Renaissance Florence that "homosexual behavior ... constituted a pervasive and integral aspect of male sexual experience, of the construction of masculine gender identity, and of forms of sociability." With this context to the bare bones of what we know about Leonardo da Vinci's sex life, it seems far more than likely that, even though he wouldn't have characterized himself as a gay man, he was participating in this world, like many other male artists of his time.

There* (Sorry about the typo)