Zermatt is a popular mountain town in the south of Switzerland, popular because of its proximity to the Matterhorn.
Cars are not allowed in Zermatt (only electrical local vehicles permitted) except during a month or so in the low season when construction and maintenance vehicles are let in. Due to the low traffic, even at that time its probably difficult hitching in.
Closest town to Zermatt is Tasch - 5 km. To Zermatt you can walk a nice trail, use a train (costs somewhere around 5 to 10 Swiss Francs) or use taxi (negotiable price). Hitching between Tasch and Visp is pretty easy. Matt Stock was waiting a while hitching at night from Visp to Tasch, but a friendly couple eventually stopped. Sign works well.
Hitching from Visp to Tasch (and onto Zermatt)
Stand near the traffic circle (roundabout) on the only road to Tasch. Theres not much room for cars to stop, but the road isn't too busy.
Hitching from Tasch to Visp
Best spot is right near the Train Station, on the road to Visp. Small town, very straight forward. Visp, not too much to keep you there but you can get a bike for 3 hours ride it around for free and return it, stay a day or two.