It seems like a pretty simple process and an easy mistake to make. So if an underage witch/wizard was Determined to get to a Destination, and they turned on the spot (with Deliberation) could they Apparate? With no practice or training? Is this an okay excuse to the Ministry?
Magic can be done non-verbally, or without a wand, but it can only extremely rarely be done unintentionally. The answers to this question seem to suggest that unintentional magic is pretty much only done by those who've never entered formal magical training. So the question of "excuses to the Ministry" wouldn't arise: the only underage wizard who might be able to accidentally Apparate would be a child of under 11, pre-Hogwarts age, and such children aren't subject to the Underage Magic legislation.
Not only that, but Apparation is DIFFICULT. It takes weeks of practice to be able to do it competently, even for seventeen-year-olds; it's not really as simple as deliberating about a destination and being determined to get there. Without training, it would probably be almost impossible to learn even if you were trying to. The chances of being able to do it accidentally are comparable to those of a monkey bashing away at a typewriter to produce Shakespeare's Hamlet.
Considering how much trouble the Sixth Years have with even doing it poorly, it seems somewhat unlikely:
Harry spun on the spot, lost his balance and nearly fell over. He was not the only one. The whole Hall was suddenly full of staggering people; Neville was flat on his back; Ernie Macmillan, on the other hand, had done a kind of pirouetting leap into his hoop and looked momentarily thrilled, until he caught sight of Dean Thomas roaring with laughter at him.
"Never mind, never mind," said Twycross dryly, who did not seem to have expected anything better. "Adjust your hoops, please, and back to your original positions..."
The second atlempt was no better than the first. The third was just as bad. Not until the fourth did anything exciting happen. There was a horrible screech of pain and everybody looked around, terrified, to see Susan Bones of Hufflepuff wobbling in her hoop with her left leg still standing five feet away where she had started.
"Splinching, or the separation of random body parts," said Wilkie Twycross dispassionately, "occurs when the mind is insufficiently determined. You must concentrate continually upon your destination, and move, without haste, but with deliberation
[A]n hour later, Susan's Splinching was still the most interesting thing that had happened. Twycross did not seem discouraged.
The Half-Blood Prince Chapter 18: "Birthday Surprises"
Note that Twycross, an expert in the field, doesn't expect them to have gotten it after an hour of practice; in fact, his entire course is twelve one-hour sessions. It doesn't seem as though it's as easy as all that.
While I don't want to completely discount the possibility of a sufficiently talented young witch or wizard doing it accidentally, the probability seems vanishingly small.