How are orcs made?

Answer 1

Tolkien himself states that nobody knows how the orcs were made, but it was not by corrupting elves. That in itself strongly suggests that Melkor secretly sang them into existence (probably along with many other evil creatures) in the song that created the universe that all the Ainur sang.

Answer 2

We can only assume that there are female orcs and that orcs reproduce sexually. The absence of female orcs in the books is probably mostly because we never encounter orcs at home but only orcs involved in the war.

Less speculatively, early in The Hobbit, there's a reference to the orc Bolb being the son of the orc Azog, which implies sexual reproduction. In The Lord of the Rings there are implications of orcs and men being crossed somehow, presumably through interbreeding which again suggests sexual reproduction. Also, Tolkien describes somewhere in his writings (sorry - maybe its the Silmarillion, maybe it's in the History of Middle-earth or any of the countless pages he never published in his lifetime) that orcs were created when Morgoth captured, tortured, and mutilated elves, procedures that somehow warped not only the individuals but also there descendants. This is part of a theme where evil cannot create life but can only twist things created by Eru, the God in Middle-earth. Morgoth couldn't create life de novo, from scratch, but he could warp life into evil forms. So, orcs presumably are physiologically rather like humans and elves and reproduce in the same fashion where life is inherited by children.