I’m not currently doing any therapy work with the beasties, but from what I recall, the means to get into it varies based on what organization you work with.
In my area, Pets on Wheels is the largest group, though my big guy was tested by a TDI person. Each group has their own test and other requirements that must be fulfilled in order to work as an official member of the group (which is important, as it generally conveys coverage under the group’s insurance policy).
In my experience, most of the tests are based on the AKC’s Canine Good Citizen test, with several additional tests added (like children, working around people using various medical equipment, like an IV pole, walker and/or wheelchair.) A “leave it” command is VERY important, as they want to make certain that Fluffy won’t go diving for a patient’s dropped meds, etc.
In your case, I’d look into your local area - contact the places that you’d like to visit, and see if a local group is already visiting them - then you can get in touch with that group directly, to see what they require of you and your pup. You can also check some of the national websites, and see what organizations are working in your area - Pets on Wheels, Therapy Dogs International (TDI) and the Delta Society/Pet Partners are just a few that I know of.