I don't have any answers and you may have already tried these ideas, but just in case I thought I'd review what I do in trying to find resources as a volunteer in local (non-HD) programs.
You are right about foundations and corporations giving to groups so you need to find the groups who have money to help the disabled. Two good starting points at the United Way and any state or regional associations of nonprofit agencies/organizations. I would check their web sites, if they have them, and also call them. One of their member agencies might be able to help.
People in state government agencies are a good source of information, too. Any agencies helping the disabled are obviously a good start. Although you are inquiring for someone who is not elderly, agencies dealing with Senior Citizens often know about resources for the disabled. Some states have an Office of the Public Guardian which acts as guardians for people who are not competent to take care of themselves and have no family or friends who are willing or able to act as guardian. Although you don't need the services of this office, I have found our state office to be very knowledgable about medical and disability resources and issues and worth a phone call.
When I start calling, I also start a note book, writing down the date, name, phone number, and organization of the person I talked to, and of course what they said. When people tell me that they don't handle the resources or programs needed or don't have the information I am looking for, I ask them to suggest whom I might call who might know more. The second person may not know either but may know who does. I just keep calling!
One key phrase you might use is 'medically needy.' Most programs have either income limits or sliding scales and it may seem as if your loved one doesn't qualify. However, some programs do recognize that even those receiving pensions or disability payments that put them over the low income capacity can be needy because their medical expenses are so high.