To answer the question:
Austria is very child-friendly.
But there are so many different things they could do:
Book flights with a cheap airline,hire a family-sized car (don't forget seats for the kids will cost extra as it will if both parents will be driving), book cheap accommodation (airbnb is NOT always the cheapest option). They can do all this themselves online but it requires a lot of homework. And then they have to find out for themselves what sort of activities are available once they get there, which means even more homework and preparation.
That would be the cheapest option.
At the other end of the scale, they could find a "family hotel" which offers full board and all-day entertainment for the children.
Depending on how old the children are, I would not recommend going too far. Travel with small children is not worth the hassle if you're just going on holiday. However, if they are all over, say, 8 years, they will learn a lot from the experience of flying and travel abroad and the journey itself can be made fun.
With our 3 we just did what we had always done as a couple: chuck tent and sleeping bags in the car and drive to France. But they stopped coming on holiday with us from about the age of 12. So your daughter should treasure the time with the children!