Liechtenstein has a few hotels, although they are generally on the pricey side. In Schaan, there is one youth hostel, but it shuts for the winter. In nearby Feldkirch, Austria, you will most likely be able to locate cheaper lodging.
The only full-service campground in the Principality is Camping Mittagspitz. It has great amenities, a pleasant welcome, and a fantastic restaurant that is fairly priced. In Liechtenstein, there are three additional campsites. Bendern, Vaduz, and Triesen each have one. All of them are pretty much filled all of the time.