You can play a dwarf mage however you want. Dwarves generally live in the lore hole, so it isn’t really described, but I’ll share my understanding of this.
The humans and elves had been sharing arcane knowledge for roughly 300 years by the time the earthen awoke as dwarves and left Uldaman. The humans and dwarves, as well as gnomes, crossed paths at some point in the last 2000 years and have been buddies since, so it’s reasonable that Dalaran may have been a source of arcane knowledge.
That being said, it’s possible that the dwarves discovered arcane (and other magics) from other sources - the gnomes are powerful spellcasters as well but they have a presence in Dalaran, dating back to Alodi iirc. The presence of golems in the Bronzebeard heritage questing implies that Dwarves retained some inclination for magical machinery, so some forms of enchantment or runesmithing might originate with the dwarves.
The Dark Irons are definitely the most magic inclined of the clans, but until about 250 years ago in lore (War of Theee Hammers) they weren’t a particularly distinct identity. The clans weren’t split back then, so it’s likely that the Dark Iron sorcerers and similar spellcasters were also spread throughout the Wildhammer and Bronzebeard clans. When the Dark Irons split and accidentally summoned an elemental lord… the other two clans probably grew a dislike for that sort of magic.
Bronzebeards probably stopped training or encouraging many wannabe mages after that, but there’s nothing to say that they actually did or that they stopped the training of every mage. They likely had mages that predated the War of Three Hammers - and Dalaran still exists as the primo mage city of the world. Bronzebeards still had some casters, even if mages were rare, in that they still had priests.
Wildhammers probably stopped training mages as well, replacing them with shaman for the most part. I’m not certain mages would’ve been popular among the Wildhammer even before the war, but magic involving lots of math and practice doesn’t fit the general themes of the Wildhammer.
And of course, when the Dark Irons returned to Ironforge they are implied to have taught more dwarves the ways of magic. So if you want a new mage, that’s an easy in.
TLDR: Dwarf mages are likely pre-War of Three Hammers veterans (250+ years old), a somewhat uncommon mage trained in Ironforge since then, a Dalaran schooled mage, or a recent convert to the Dark Iron ways of spellcasting.