There are organizations all over the U.S. and world that offer financial assistance to people who are awaiting organ transplants, have already received them, are trying to become live donors, or have already donated. Some are niche-focused, helping only transplant recipients in a certain state, for example. Others are much more broad, providing financial assistance for health and living expenses to donors and/or recipients before, during, and after surgery. Recipients and donors can even qualify for special academic scholarships, like this one.
Oh, I wish we'd known this sooner! We're looking into fundraising now to help Joe pay his medical expenses and recover from the last six months of downtown Chicago living with no income whatsoever. The deeper we dig, the more we realize we lost out by not taking advantage of these opportunities sooner.
I've added several links to organizations that offer financial assistance, in the right-side navigation bar under "Resources for donors & recipients." To find others, try Googling "transplant assistance" or "organ transplant foundation" etc. Add your state or city name to the search to narrow the results -- you may find a small local fund that could help.