In general, youth programs must make “reasonable modifications” to their programs to accommodate a child’s disability, but there are exceptions. How the rule and exceptions play out depend entirely on the specifics of your program and a child's disability. You will have to assess each situation individually. Two cases -- chronic illness and communicable illness -- illustrate how you might apply these principles.
Children with chronic medical conditions require a nuanced calculation. Most organizations can accommodate food or environmental allergies, for example, with moderate changes in their programs. Children with compromised immune systems will need more proactive plans, such as added protection from children with colds or flu symptoms.
Diabetes is another common situation that caregivers face. Because the treatment involves sharp, pointy objects, most youth programs are not comfortable dealing with it. Lay people, however, are perfectly capable of learning to monitor blood sugar and dispense medicine (even insulin), so the ADA requires that you provide those services. Note that the American Diabetes Association encourages older children to learn self-care and urges programs to allow them to do so. This guidance flies in the face of most "nurse-administered only" policies, and may require you to adjust that policy in your program.
Other illnesses may require more specialized medical care. The law does not require you to perform medical procedures, or to hire a nurse to provide advanced care. If a lay person cannot be trained to safely provide a service, then your program need not provide it.
Most illnesses, however, simply require you to train your staff to recognize symptoms that require more medical care, and to have a protocol for getting the child to the doctor. For example, a child subject to seizures can be accommodated in most programs simply by training staff on (a) how to respond during the seizure, (b) what procedure to follow in alerting parents, and (c) when to call for emergency medical care.
You can exclude children who pose a direct threat of passing on a communicable disease, such as measles. CDC and health department rules around the coronavirus have operated on that principle. Some communicable diseases, however, you can accommodate. For example, Methicillin Resistant Staphylococcus Aureus (MRSA) is contagious, but the CDC counsels against excluding children from day care or schools if you can keep the infected area clean and covered.
Every program that serves young people wants to do the right thing for them. Both morality and the law require that we include children of all abilities in our programs. But even federal law must bow to the laws of physics and economics, and only require that your program do what it reasonably can to accommodate disabilities.