I am disabled so I am also sometimes in this situation. First, disability and chronic illness does count as "diversity." But when applying for fellowships, you need to pay attention to the specific language of each individual fellowship. Some fellowships might specify that they want to support students of color. Others might specify that they want to support first-generation students. Still more might just ask for anyone with a "diverse perspective" or other generic language. If the fellowship falls into the latter category, then yes, you are eligible. You can make a case for yourself in the application materials. And if you really aren't sure, just reach out to the granting institution and ask!