I won't call this an actual answer, but I would like to call attention to one aspect of Tolkien's Legendarium and that is that in it, magic works a little differently than what we're used to seeing in fantasy stories. "Magic" or sorcery as we would think of it is typically reserved for dark forces, whereas the magic of elves, dwarves, and even hobbits (I'll get to that) is more akin to a deep fundamental knowledge of crafts, in ways that, to us, would seem magical but really just involve a level of skill and understanding that we can't really achieve.
Examples of this being Gandalf's ability to set things alight, but only in the presence of a flammable medium - he explicitly states in Fellowship of the Ring when they are on Caradhras that he cannot simply create fire. Another example being in concern to the "magical" properties of the elven cloaks given to the Fellowship by the Galadhrim - when asked if they are indeed magic, an elf (maybe Galadriel, I'm sorry I don't have that source material at hand at the moment) responds with an explanation that is basically the gist of my whole response here.
So, concerning the magic of dwarves and even hobbits, I believe it is along these same lines. It is explicitly stated in The Hobbit, that hobbits have "little or no magic about them, except the ordinary everyday sort which helps them to disappear quietly and quickly..." (The Hobbit, An Unexpected Party). As The Hobbit is supposed to have been written by Bilbo, this implies that, to the races themselves, the magic they possess is simply not considered really magical. As far as Hobbits are concerned, their ability to go about silently is just a skill or talent, apparently also possessed, to a lesser degree, by dwarves. But for 13 dwarves and a hobbit to be able to shuffle past and yet "I don't suppose you or I would have noticed anything at all on a windy night, not if the whole cavalcade had passed two feet off" (The Hobbit, Roast Mutton) is definitely what would be considered, by our standards, a magical ability. Try walking alone, through the woods filled with grass and sticks and dead leaves 20 feet from someone without them noticing, let alone 2 feet away accompanied by 13 others.
In summary, I think all races in Middle Earth have magic, to varying degrees, at least by our standards and understanding, and all members of those races are at least capable of learning said "magic". Even Aragorn, a man, has obviously "magical" healing abilities, and possibly magical combat abilities (as he was able to make it through an entire war without a single scratch, and strike terror in his foes with his presence alone).
It's also possible that, as The Hobbit was originally a stand alone story, nothing I've said is relevant and the dwarves really were casting outright supernatural magical spells.