My current sewing group supports several options. They mostly make receiving blankets and burp cloths from flannel. They also make totes.
It is my understanding that blankets/quilts aren't recommended for infants until over 1 year. Many organizations for things like premie or medical comfort blankets have very strict fabric (and thread, batting) requirements as well as size.
I like larger projects anyway, so I look for foster kid and "transitional housing" (typically families fleeing domestic abuse). I prefer organizations that let the recipients choose what they want -- you could be delighted with one of my projects while being dismayed with another.
PS -- a lot of animal shelters don't want projects filled with fabric trims either. They don't wash well.