Often use Municipals in France and wondered if there was an equivalent in Italy?
We've just come back and used several aree attreziata - a cross between an aire and a municipal campsite though all privately owned. Some allowed caravans but they were mainly used by motorhomes.
Prices were around 12-18 euros per night and for that you have electricity, showers, laundry facilities, washing up, the usual servicing facilities for the van and sometimes a recreation /TV/internet room.
The only difference between these and campsites is that there is not usually much grass around and they are more like large car parks with some grass.
Try those at Lucca ( very easy to get to Pisa from there on the bus- about 30 minutes on a cheap, aircon coach which delivers you right to the entrance to the Campo dei Miracoli ), Orvieto ( again, very convenient for the town as it is at the bottom of the funicular) and Modena
If you look in the MHF database several of them are there and there are lots of databases of sostas and campsites in Italy on the internet. We did not find many sostas where we felt comfortable and rejected several. They tend to be in the municipal car park and- bluntly- distinctly scruffy and rough.
We stayed at several small sites around the Ligurian coast and towards France. I'd also recommend the Campeggio Citta di Milano where your children might enjoy the small zoo and the huge waterpark next door ( discounts for campers) See my review of it in the MHF database ( ie press the blue "Campsites" under my avatar on the left)