Exceptional medical and social reasons have to relate to the need for one particular school.
So yes, it is possible that being a Young Carer could come under the category of exceptional social need if the location of the school meant that the child would get home from school much more quickly than any other school they could be sent to or if the school chosen offered specialist support and intervention for Young Carers that no other local school could offer.
However being a Young Carer (or in the medical and social category for any other reason) does not mean being allowed to have a free pick of all schools. It just means that of a child has a very special reason for attending one school above all others, it can be taken into account.
The other thing to note is that you need to declare (and prove) qualification for this category at the time of application. Special circumstances can be used at appeal but you cannot argue the council was unreasonable not to consider these needs if thwy weren't told about them.
Sorry - I see it is an appeal not an application. IN that case my last point applies. If you told the council about your child's status as a Young Carer and offered evidence about why the school you asked for was the only one suited to their needs, then you can appeal to say that the council was so unreasonable to ignore this that the decision should be overturned. That applies to ICS appeals which are harder to win.
For class sizes under 30 or children older than Year 3 appeals, are easier to win and evidence about why one school meets the child's needs best would be potential grounds to win. You wouldn't have to also show that the council had made an "unreasonable" decision to deny a place in the first place.
Practical experience of being on panels would say that panel members would be as sympathetic as is possible to a young carer as they do recognise the stress and extra work that this puts on what is a very young person. There is never a guarantee of success but I would definitely go to an appeal pushing the need for a specific school because they are a young carer. But they will need to be assured that this is genuine, so can you produce written evidence that they are a young carer.