Do Special Attack Pokemon deal special attack damage with their basic attacks?
No, but it's more complicated than that.
All basic attacks deal normal damage, so far as we've been able to verify through testing. However, each Pokemon's boosted attack will deal increased normal attack by items that match each Pokemon's hidden damage type. For instance, Alolan Ninetales will deal increased damage with its boosted attack when you add a Sp. Atk item to it - but Cinderace will not. Inversely, Cinderace will deal increased boosted attack damage when you add an Attack item to it, and Ninetales will not. See our Pokemon Guides to learn which stats impact the boosted attack of each Pokemon.