HarrysMummy17 and Jessica I agree with iwishkidslikedtomatoes
I think it is really best to ask the adoption agencies about these things since they will know their own criteria and it may vary. You can try any number of local agencies (local authority or county council or voluntary).
I think being able to see the child into adulthood is something that they look for so it really depends how serious an illness is. Also you need to care for the child, either you/your friend or your partner, if you have one, so if a medical condition meant it would be hard to parent a young child that may limit things.
Just for the record I have high blood pressure and a few medical conditions which are totally managed by medication. These were accessed in my medical. I am certainly not a stunning example of fine physical health (a few people on here have met me in person so can attest to that!) but I passed my medical OK and went on to adopt and was already parenting a child.
Do ask the questions to the right people. Awful to want to do it and not because of lack of asking. And if the door is barred to adoption, consider fostering because the rules might be slightly different. You could still care for a young child or perhaps an older child on a semi permanent basis, and would get financial support and other forms of support.
When I first became a parent it was all about having 'a baby' and I looked at friends with much older kids and thought it must be strange to 'parent' an older child. Now I have an older child (10) who I adore I realise that all the ages and stages have their benefits and delights (and heart breaks) and to miss out on all of it because you cannot do the initial bit is very sad. My son was 3 when he came to us and I was sad to miss out on the first few years of his life but soon he will have been with us longer than he has been anywhere else (well, not soon, but in a while) and I realise that the joy of knowing him (he is pretty gorgeous) is far greater than my fleeting early day thoughts of wanting to adopt a baby or a toddler.
I hope that helps, it sounds a bit smug, but I do not mean it. I just mean that if there are any limitations on a prospective adopters abilities then adopting a slightly older child, who needs a bit less hands on care, or long term fostering, might be a better option. But please do discuss all options.
All the best.