Leichtenstein is a tiny country, consisting of 160 square km or 62 square miles. As such, while it will have its own population of chamois and Ibex, any population of the large predators is shared with countries around it.
Ineed, it is not thought that any of these animals have set up territories within Leichtenstein, though they doubtless come through on occasion.
The majority of Leichtenstein economy is related to finance, which it has significant numbers of organisations to service.
With healthy populations of Ibex and marmots, as well as foxes and badgers and other smaller mammals, it is never the less to small to have any of the big 3 living permanently within its borders. A wolf pack could easily have a territory of several hundred miles, which in this case would include more than the entire country.
However as the populations of these large predators grow in the rest of the Alps, they are likely to appear here more regularly. Both bears and Lynx could eventually make this small country home.
In any event, with much of the country available for wildlife, any wild oriented holiday can only help improve the outlook of these animals. Hiking in the mountains are still likely to be rewarded with sightings of deer, chamois, possibly ibex and a range of other animals. Always make clear you are interested in the wildlife of the area when you visit, this will make it more likely that the animals will stay.
Below is a map of places to stay. Being a small country there are not that many to choose from, though it is worth remembering that staying across the border and driving in for the day – or indeed several days will not take up much of your time.