How To Get Started With Hand Cycling
Feel free to edit this answer to add more resources.

Hand cycling is a Paralympic sport, administered by the UCI. There are links at the bottom of the UCI page for all world regions.

Here is a link for Disabled Sports USA, describing different kinds of bikes and listing local chapters and a bunch of useful links.

The United States Handcycle Federation seems to be mainly about competition.

Cycling Australia has a Para-Cycling page, there is the Queensland Sporting Wheelies and Disabled Association, and Wheelchair Sport NSW. I'm sure there are bodies in each state, but a quick search didn't locate them.

Good luck!