Yes, I think that it's incumbent on the person with the disorder to speak up. Sleeping isn't an involuntary response to sitting in a car for most people, so if you're the one who's out of the ordinary of what people would expect then it's up to you to speak up. And if it's genuinely a disorder, then I think most people would be a lot more understanding than someone just falling asleep all the time. One is a medical issue, the other conveys that the group isn't interesting enough to keep you awake. Totally different issues.
Well, what I am questioning here, really, is "how out of the ordinary is it"? Sure, I personally fall asleep very easily even while medicated and I have a reason (though anyone I am close enough to to be driving that far with, probably already knows this) but what about in the less extreme areas of this? I am sure there are many people, especially when pushed, as by a strenuous trip as described in the OP, fall asleep involuntarily. How often does this have to have happened to a person before it becomes their responsibility to say that they fall asleep in cars and to ask if it is okay, when offered a ride?
Whether it is rude or not, it seems to be very very common. Is it more common, or less common (the tendency to fall asleep in a car, even if it's not extreme) than the tendency to be annoyed, as a driver, by this, or even to be in a position where sleeping passengers is a safety issue? From my perspective I assumed the sleeping is more common, but I could be wrong, but I ask because you are correct, the one doing the "out of the ordinary" thing, rude or not, is the one who should speak up.
And it gets back to one of my questions: How do you politely decline the offer of a ride if you can't or won't abide by the rules of the owner of the car? It's very generous of someone to offer to drive the whole way and not ask for money for gas. How do you say "thanks but no thanks" without being insulting? And if it's a disorder and not something you can control, then if sleeping passengers causes a safety hazard to the driver, it doesn't matter if you can control it or not-it's still a safety issue. That would be another tricky area of politeness, I think.
But since this has happened several times to the OP and she's obviously aware of how the rest of the group feels, then it's up to her to come up with an alternative if the rest of the group's preference isn't okay with her.
If this really is the case, then again, what is the best way to turn down what otherwise seems like a generous offer? And if she isn't absolutely sure that she will fall asleep, is it worth the social consequences of looking rude by turning down that generous offer, or even of losing what time with her friends she would be sure to enjoy?
There are polite and impolite ways of handling anything, of course, and I think it would be good to discuss how to handle this politely.