Can I ask why you want them to be 100% cotton, and also why you're giving quilts rather than blankets or sleeping bags? From what I know, something like a quilt is more likely to be stolen in that situation. Polyester is warmer than cotton, and also much lighter. Cotton takes a long time to dry after getting wet, too. There's a reason why sleeping bags aren't made of cotton. I started thinking about ways to make an all-cotton quilt warmer without having to spend a fortune on deluxe cotton batting, such as using corduroy for the quilt back, but that has the same drawbacks: heavy and takes longer to dry.
If you are aiming to help people who are sleeping rough outdoors, I suspect quilts aren't a good option. Have you spoken to charities who work directly with unhoused people, and better still, to people who have lived experience of this? Why not donate items they really need, or simply money? Perhaps make a bunch of quilts to raffle off and send them the proceeds?
Alternatively, if you want to send quilts to people in need, there are lots of folk in poverty who have roofs over their head. I've been homeless, but I was "hidden homeless", which is really common. I was sleeping on friend's floors, and then in council homeless accommodation, and eventually was put into a long-term council flat. I was very lucky that the council homeless accommodation was a one bedroom flat, as a lot of folks are in hostels and such. Anyway, the point at which a quilt would have been something I could use was once I had a reasonably stable roof over my head and was no longer packing up a bag and moving around every few days or weeks.
A lot of people are having to choose between heating and eating these days. Disabled people are particularly at risk. I've just made a quilt in a week (this is more of a big deal because I'm disabled and only sew by hand) for a friend who has the same disabilities as me, can't afford to heat her home properly, has terrible circulation, and was shivering despite four layers of clothing and the bedding piled up. So I made her the warmest quilt I could, which meant needlecord both sides, plus a bit of velvet in the top, cotton velvet is surprisingly cheap, and polyester batting. The only thing I bought was the batting, the rest was stash.