I want to start by making it clear that I am not endorsing any non profit company as I haven't talked to any of them I am sure there may be more but I am showing a few companies that I have found using a Google search of Autism Service Dogs. Also I want to note that not all of them work with young children.
I am posting the 1st 3 I found, I included links so you can look if you want more info about thees places. If you have any questions let me know.
Canine Companions for Independence
This company says the dog is free but you have to travel there, pay for hotel and food. They have offices in Colorado and Chicago. I am guessing that is where you would have to travel to.
Dogs for the Deaf
With them there is no fee other than a $50.00 application fee and a $500.00 good faith deposit that they say you get back after 1 year. They are located in Oregon, I couldn't fin any other location. One thing I found interesting is that they say the dogs are not trained to go everywhere with the child such as the mall, park zoo etc. My question is what's the point of goign through the process if the dog has limitations. As a mom to a child with Autism the dog must be able to go everywhere with the child because the child needs the dog in various situations otherwise you might as well go with another company.