16a. Anyone who requests a copy of the game may have it for their video library.

16b. Anyone who requests a copy of the game may have it for his or her video library.
What type of pronoun is this?

same as the later post.

The pronoun used in both 16a and 16b is a reflexive pronoun. In this case, the pronoun "it" refers back to the game that is being requested. By using a reflexive pronoun, the sentence avoids repeating "the game" again and again. In 16a, "it" is used as a gender-neutral pronoun, while in 16b, "his or her" is used to show inclusivity towards both genders.