What are the consequences for someone who swore on the Qur’an that they would not engage in sexual acts with anyone, and then broke their oath by committing an act of sodomy (without penetration), and how can they atone for their oath and have their sin forgiven?

The abomination of sodomy is among the major sins, and its prohibition is well-known. An oath to abandon it is binding, and if it is sworn upon the Qur'an, it is even more emphatic. Committing what you swore to abstain from makes you sinful and a breaker of your oath, even if no penetration occurred, for the intention is what matters.

You must promptly offer sincere repentance and resolve not to return to the sin, for Allah accepts repentance and forgives sins. You must also perform expiation for the oath: feeding ten needy persons, or clothing them, or emancipating a slave. If you are unable, then fasting for three days.