Rifugi Bivacchi provides lots of maps. As you can see see there are a lot more refuges than bivvy shelters, but refuges are not free. When I spend time in the italian Alps, using refuges a lot, my budget is 50-60 Eur per day. I take it that's not what you're looking for.
Still, there are bivvies here and there. When you're planning to go up there, ask beforehand about the state they are in. And check the accessibility:
Bivacco Giuliani for example is on an (easy) rock climbing route, so unless you're a climber, you can't get there. For hikers, something like
Bivacco della Pace looks more suitable.
I don't have a car myself, and rely primarily on public transport and if that doesn't work on hitch hiking. In my opinion public transport in Italy is pretty cheap. There are no buses heading up into the mountains real early, but apart from that, there are buses all over the place, over all the mountain passes and consequently to all the trailheads. My main problem is usually by the end of the day. The last bus out of the mountains often leaves as early as 6 pm or so, while there is still daylight for several hours. If the weather is good, I like to be out there the whole day. Considering the bus schedules, Italians apparently don't do that, nor do they expect tourists to. My solution? Hitch hiking. Rarely fails, though I have to admit that I also have had to hike down a mountain for several hours. And I've always got some cash with me to pay for accommodation and food if I don't get back down to where I want to go and have to spend the night in a refuge, hotel or whatever I can find.
For the record, I'm hitch hiking as a 56 year old guy, alone, and I usually look a bit scruffy when I'm in the mountains. I like to think that I portray a harmless kind of scruffiness, but I still don't think it's conducive to getting I ride. However, what works in my favor is that my pack and hiking poles make me readily recognizable as a hiker, which means other hikers tend to be sympathetic to my plight. It rarely takes long to get picked up. Sometimes even single young women stop for me so I guess I look harmless. The main issue is that around dusk there is hardly any traffic left in the mountains, so I aim to be by a road while there is some daylight left.
Outside the mountains I find hitch hiking in Italy much more difficult. Lots of cars, but it can take hours to get a ride. If a train or a bus is going my way, I'll take them instead anytime.
July is fine. For Italians, the first half of August is the most popular time to head into the mountains. There will be some weekend warriors from July 22 to 24, but you're still ahead of the big rush.
Cheers, Rob