Is it sinful for me to refuse to see my stepfather after he cheated on my mentally ill mother?

The Islamic prohibition on cutting relations with relatives only applies to blood relatives, not in-laws, stepparents and children, so you are free how you decide to treat him. It is of course best to forgive him if they are still together. Islam also forbids not speaking to a fellow Muslim for more than three days, if you are both Muslims you can try to maintain a polite relationship using your phone and social media while avoiding him in person.