Is there a word for a person who doesn't masturbate? If so, what would he/she/it be known as?
You could call such a person an onanophobe.
asexual --> an-onanist?
Which is to say, I do not know either, but other people are attempting to coin a word that is the opposite of an onanist--
a man guilty of the sin of onanism, or spilling his seed. This is an allusion to Onan, in Genesis 38:
Gen 38:8-10 (NRSV)
Then Judah said to Onan, ‘Go in to your brother’s wife and perform the duty of a brother-in-law to her; raise up offspring for your brother.’ But since Onan knew that the offspring would not be his, he spilled his semen on the ground whenever he went in to his brother’s wife, so that he would not give offspring to his brother. What he did was displeasing in the sight of the Lord, and he put him to death also.
While Ronald's aversion to onanophobe is well reasoned, it does have the benefit of a literary reference.
Some people who don't masturbate are anhedonic.
ANHEDONIA : a psychological condition characterized by inability to experience pleasure in normally pleasurable acts — an·he·don·ic adjective
Sexual anhedonia, or pleasure dissociative orgasmic disorder is a real thing:
a condition in which an individual cannot feel pleasure from an orgasm. It is thought to be a variant of hypoactive sexual desire disorder.