Is there a sin upon one who swore not to marry if he does not marry, especially if the reason for his unwillingness to marry is extreme shyness toward women, and he seeks to eliminate sexual desire?

Modesty is a praiseworthy quality and a part of faith, but it should not prevent one from performing what the Shari'ah has commanded, such as marriage. The Prophet (peace be upon him) encouraged marriage and forbade celibacy. He was the most modest of people, yet he married. It is not permissible to completely eradicate one's sexual desire; an oath to abandon marriage is an oath to abandon a Sunnah and a good deed. Therefore, an expiation for an oath (kaffarat yamin) is due for whoever swore to abandon marriage, and then he should marry whenever it becomes feasible for him. The ruling on marriage varies according to a person's situation. You should be patient and supplicate, and consult those with experience.