Yeah, it's definitively possible to camp, and it's what I did to save some money. You can call for rates, but it'll be around 20 swiss francs per night for campgrounds.
I'd bring your backpacking tent just to save weight and space when you will be lugging it around (are you taking trains or car?). And also if you want to backpack and "wild camp" somewhere you'll have that option and can save cash for a night.
In Zermatt there is one campground.
You get out of the train station and walk left, back towards Tasch, and the campground is close to the train tracks on the left side of the road to Tasch.
You can't drive to Zermatt (there is a road, but not public) - Tasch is the nearest town to Zermatt where you can take a train from. I wouldn't base camp out of there if you want to do stuff in Zermatt, but if you get to the area late you can stay there for the night and take the train to Zermatt the next day. The campground in Tasch is located at the end of the town - basically one of the last "things" on the road from Tasch to Zermatt - it's on the right side of the road.
As for camping in the Jungfrau region - I'm sure there are tons of campgrounds in Interlaken. I stayed in an awesome campground in Lauterbrunnen, right outside Interlaken, and a short train ride away from Wengen, Klein Scheidegg, Grindenwald, and the Jungfraujoch (and the Eiger).
Lauterbrunnen is small, beautiful, quiet, private, and nice- Interlaken is bigger and more of a city - I was there with a girlfriend, so the former was better.
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