I am acutally a doctor (pediatrician) as well as a person with a disabilty myself, but it is unethical to give medical advice over the internet.
However I can give medical information, although I don't know if it will solve either your cycle problem or your contraception problem.
Forms of birth control that regulare cycles are all hormonal. There are two main kinds - combined and progesterone-only. The combined estrogen/progesterone are the pill, the patch and the ring, and they all have that risk of blood clots which is higher in women who are not very mobile. Some gynecologists will prescribe combined HBC to women who use wheelchairs but who don't have other high risk factors for blood clots (like smoking or family history.) Pregnancy itself can also cause blood clots so it is a risk-benefit situation. The patch has higher hormone levels than most pills currently in use.
Progesterone-only includes progesterone-only pills and the the Depo-provera shot once every 3 months. I believe there is also a new 5 year implant that is progesterone-based. Many women on progesterone-only methods stop having cycles entirely after a couple of months.
An IUD may or may not help with the cycle problem (the Mirena might because it contains a small amount of hormones) but it sounds like those are not options.
Other forms of contraception that do not involve hormones at all are the male and female condoms, and the diaphragm, with spermacide.
This is all information your gynecologist should know much more about than I do and be willing to work with you on as much as possible. There doesn't seem to be one obvious answer so you and she are going to need to have some discussions and try to work out something that is safe and will work for you.