To put it bluntly buddy, if there is no court order or anything legally established, the person who has custody of the child at the time is the primary custodial parent.
In another word, possession of the child is 9/10th of the law. Which in your case means that the person who has physical custody of the child can dictate what the other parent can and cannot do, what they need to do and what not to do to see the child.
If, in your case, the other parents decide that if you want to see the child, you need to not only go there to get the child but you have to drive there in your own personal vehicle and you have to be there in 6 hours or else, then you pretty much have to drive there in your own personal vehicle and hope that you get there in 6 hours or less.
The current custodial parent can denied the other parent visitation, limit visitation, or just about anything that they want to do.
And you know what, the non-custodial parent cannot do anything about the situation. They only way force the custodial parent for custody or visitation is to hire a lawyer and take them to court and sue them for the right.
So, the answer to your question is simple. Legally speaking, the non-custodial parent has NO responsibility for doing anything not allowed by the custodial parent. If they want you to go and pick up your child 300 miles away, you are going to have to do it if you want to see them. Of course, once you have the child, you can turn around and do the same too.
Then again, like the first poster said, you guys can come to an agreement and meet half way. But, unless you go to court and get an order stating that, it is not legally binding to either party.
Well they probably have restrictions on who they can give it to based on the nature of the disability if it's an organization that's meant to help people with certain disabilities. You could certainly try pursuing legal action- with a good lawyer you could probably win this case- but short of that there's nothing.