If I sit for hours every day working, can my varicose veins get worse?

No. Fortunately, sitting doesn't cause you to get varicose veins and doesn't make them worse, as dr. Bulger said. Standing is actually far harder on veins than sitting is. Sitting for long periods of time can make symptoms from varicose veins worse though, especially swelling. Getting up and taking 5 minute walks every hour or two can make a big difference.
No. Sitting in and of itself does not lead to venous insufficiency and subsequent varicose veins. Prolonged immobility may lead to venous stasis and formation of thrombus, which can damage vein valves. Unmoving for long periods such 8 hours without standing or moving the calf muscle, and additional risks such as obesity might be present to truly increase risk. Best to get up and walk around.