How can I legally get permission from the foundation Autism Speaks to name my website after my autistic son "Corey Speaks"?

You could contact the other larger organization, and give them your thoughts, and you could also hire an attorney to do that for you, preferable if you can afford it. By the way, having worked in and around school districts for the better part of the last 30 years, I have seen many children on the autistic spectrum, and applaud you in your motivation. My observations have been of the difficulty children have in securing an appropriate educational environment - because the appropriate ones are the costly ones. I sincerely hope you can work something out, or come up with another name that satisfies your needs. It is probably wise to resolve these things up front as you notice here.

Your question involves copyright and trademark law. I have reposted your question there.

The best thing to do is reach out to an authorized individual at Autism Speaks and request permission. The documentation required to obtain that permission should be completed by an attorney so you do not have any issues. Generally you will need a license or a co-existence and consent agreement. That should always be done by an experienced intellectual property attorney.

You need to have your own intellectual property attorney reach out to the Autism Speaks organization. I have serious doubt that the law requires that you get permission from that organization before publishing a "Cory Speaks" website that's focused on providing autism information. In short, the latter may very well NOT be an infringement of the former [especially if the appearance of your website is substantially different, a proper disclaimer is published, and the content does not reasonably leave the impression your site is sponsored or endorsed by or is related to the Autism Speaks organization]. Your own attorney should know how to draft that correspondence without undermining your legal position should the organization object.