We went on a 4 week road trip last year over a lot of Northern Italy and French Alps.
It was September when we went and had no issues in finding campsites really. a few were a bit manky but on the whole, good. I enjoyed it around Lake Como and around Val di Sole areas, also Tuscany near Sienna.
The sites are on the expensive side, and average of €20-30 pn in Italy, where as it was more like €10-€15 in France. We prefer very quiet sites, away from everyone else and kids, a lot of the sites were quite big and regimented, especially around the lakes. The Diesel is also really expensive in Italy generally, €1.80-90 or so a litre compared to France's €1.45.
We had no security issues at all despite our side window breaking on the motorway on the way down and having to do a fix with polythene and parcel tape.
That Camping Dolomiti site in Val di Sole is a nice looking site, I spotted it when we were there, but a bit to Camping and Caravan club for our tastes. We stayed further up the Valley at the foot of the Passe de Tonale on a very quite little site in the village. Don't be planning on finding cheap sites near Lake Garda, very expensive IMO, nice, but expensive.