What kind of doctor is familiar with sex addiction?

No one really. A psychiatrist, psychologist, or related therapist. In a pinch, addition doc. However, no specialty really systematically addresses this problem and you must ask the individual provider specifically if he can help.
Sex addiction. Some treatment programs deal with food addictions, gambling and sex or pornographic addiction. They use similar treatment processes as do those used for drug and alcohol, sex therapists may also deal with sexual addiction in an outpatient setting. Medication use would be limited to program led by a trained psychiatrist or physician.