In reply to Kean:
Haha, here I am!
Campsites in South Tirol, I.e. val Gardena are basically non existent. In Alta badia there is one, Camping Colfosco.
In val di Fassa on the Trentino side there are three, one in Canazei, one in Campitello and one in Pozza di Fassa. The one in Canazei and Campitello are IMO not super nice as they are basically designed for hard standing requirements. Good luck with your pegs. The one in Pozza is up a hidden valley, has grass etc. It's nice enough.
As Kean says, on the far side of Passo Fedaia there is Camping Marmolada, which is in a really beautiful location. It is however expensive. But the advantage is that there are less people in general in that valley. Less facilities for sure too. Moving east, there are two, one on either side of Passo Staulanza. Both are apparently nice. And one just south of Alleghe, again mainly hard standing.
Early July is a little early in the holiday season, most people in Iraly haven't yet kicked out. The hordes really arrive in mid July and August. That said, if you want quieter mountains, the Veneto Dolomites are in general much much quieter than South Tirol and Fassa which I am sad to say have become increasingly like a circus over the last few years. Case in point, try getting parked at passo Sella, or driving back at the end of the day from the pass to Canazei. I've sat in traffic for 45 minutes to just get ti the round about at the bottom of the hill. Turning off to get to my home valley, the traffic disappears immediately... the southern and Eastern areas are perhaps less well publicised but the mountains are often bigger and more severe, so less well suited for "tourists". Mountains/ranges like Civetta, Moiazza, Pale di San Lucano, Pale di San Martino, the Lagorai, Pelmo, Lastoni di Formin, Tofana are all impressive and less busy...