Mountain holidays have heaps of things for kids to do.
Zell am See (mentioned above) has a lake and several outdoor lidos, cable cars, an alpine coaster at Kaprun, a snow play park on the Kitzsteinhorn glacier, a veteran steam train every Thursday and much more.
Nearby Saalbach-Hinterglemm doesn't have a lake, but it has a great outdoor lido, cable cars, Europe's largest high ropes park (like Go Ape) with areas that even a 6 year old can do, treasure trails on the mountains, numerous playgrounds at the top of the mountains, mountain restaurants, mountain biking, mini golf, archery.
Both resorts offer summer guest cards that give free use of the cable cars and most outdoor facilities, and both are in Salzburgerland, where public transport - trains and buses - are free of charge to tourists who obtain the Mobility Card from their accommodation provider or tour operator.
If keeping the kids occupied is a concern (there's already plenty to do!) there are also hotels that cater for families with children. There are marketed in Austria under the common term of "Kinderhotel" although each hotel is independently owned and managed, usually by a family. These hotels have pool areas, play areas, usually a nanny or a team who entertain the kids, and often also a working farm with animals to pet and feed, ponies to ride and so on. Google "Kinderhotel Austria" or look at somewhere typical such as the Ellmauhof in Hinterglemm for more details.