If a person who has never been married is party to committing adultery with someone who is married, does that make both parties adulterers and disqualify the unmarried person from ever marrying?

Adultery (moichao) is defined as having unlawful intercourse with a married person.  Fornication (porneia) is defined as unlawful sexual intercourse in general, which would include adultery, pre-marital sex, homosexuality, etc.

What makes the sexual intercourse defined as adultery is the fact that one of the participants is married to someone else.  If that were not the case, then both of them would be committing the sin of fornication which is sex outside of marriage.  As it is, both are adulterers in one sense in that both are committing the sin of adultery due to one of them being married.  Yet at the same time, it could be said that the person committing the adultery is the one who is married.  The other one is committing fornication by having sex outside of marriage.

Jesus’ commandment in Matthew 19:9 addresses two people who are married.  If either one of them divorces because their spouse committed fornication (by definition, adultery), then they are allowed to remarry without their remarriage being defined as adultery by God.  Their spouse whom they had divorced because the spouse had committed the sin of adultery against them would not be allowed to remarry.

What Matthew 19:9 does not cover is the one who has never been married in the first place.  In your scenario, it would be the unmarried person who is having the affair with the married person.  One of the reasons for getting married is to avoid fornication (1 Cor. 7:2, 9).  If this person repented of the affair he or she was having with the married person and later on found someone else and wanted to commit to them via marriage, they would be scripturally allowed to do so.