Goodness me I found this requirement really tricky,, especially with the younger children.
I've not got a mass of solutions, but could you speak with someone at a religious institution (Sunday school type thing) to see whether they might help you get some experience? If you are in London, children's centres with home start offer a buddy volunteering system where volunteers are paired with struggling parents to offer support a few times a week, that could work. They need people to help mums with the after schools and evening sessions.They also need help with daytime sessions, although I get that isn't very practical. You could also consider asking at your local nursery if they specifically want someone to come in and read with the children of a younger age, although you might have more truck with this at a primary school level. You could become a governor of a local school, or volunteer for the PTA of the local school or a friend or relatives child's school. I'm sure they would love some volunteers at the local school fair, and you would be interacting with the children there. Could you read stories at your local library? Volunteer at their Lego club? At a push, if you have a while and a desire for a lifestyle change, you could register as a childminder? You could look and see whether adoption UK had any suggestions or advice too. Bigger hospitals also sometimes have volunteer play people in their children's department or kids wards to help kids who are nervous engage with the toys and stuff in the waiting room, could that be an option once a week?
In terms of babysitting, could you volunteer to go out with parents for a day and let them supervise their kids in the park whilst they go for a wander for an hour? Free, and then you have that ticked off as you were in sole charge of them. You could build it up as you go along. Or even just asking friends to be present with them through the night time routine should work- ok, so it wasn't you putting the baby to bed on your own, but you saw how the comforted their child who had a nightmare or wouldn't go to bed. We spent a lot of time going to the zoo and crashing friends family days out.
You have said you have reached out to summer camps, could you try something with a specific interest of disabled children, like a riding school?
You could also look and see whether rainbows/squirrels need some volunteers.
Just on another note, it's particularly tricky to get the type of experience needed because for example I know for sure I couldn't inflict my two children on a well-meaning adopter to be mainly because I still want them to consider adoption afterwards 🤣 but for real, attachment issues are no joke, so actually getting first hand experience of direct care for kids like the ones you might be placed with is actually really tricky because their caregivers obviously work quite hard to protect the bond. It's one thing the child going to nursery, it's another one entirely to go be taken care of by someone who isn't a well known adult! So I get it feels frustrating.
The recommendation to get some experience with disabled children often focuses on children with severe physical impairments, but from my (limited) experience, for most children who are placed for adoption, that is not their defining or most prominent need. It is about attachment, and (if I can get on my soap box for a moment) I get frustrated that there is this obligation to prove you can build a relationship with a child and commit to volunteering if you will then most likely have to stop to commit to your own child. And if the volunteering doesn't rely on relationship building, how does it build the skills for adoption? The closest thing I can think of that might actually be useful might be an autism group as the behaviours of developmental trauma and autism overlap somewhat, although they have very different needs. Or maybe an fasd group if that exists.
Anyway, I hope some of these ideas might be helpful!