The people who are qualified to treat varicose veins are those who specialise in venous disease, although it is best to see somebody who is a vein (venous) specialist: not just a vascular specialist.
Most patients are referred to see a Vascular Specialist rather than a Venous Specialist. These Venous Specialists can either be Vascular Surgeons or Interventional Radiologists. It doesn’t matter who you see but it is important to see somebody who’s got a special interest in vein disease.