Well, one way to say if something is viable as AD or AP or not is to look at the abilities it has objectively before jumping to a conclusion on if it would work or not.
Ashe has one ability that scales off AD, her volley and her ultimate for stunning people and initiation, however she only gets scaling damage on her ulti with AP. She does however have an auto attack modifer that lets her slow people.
Veigar has a stun that has a much shorter cooldown and a potential to stun the entire team, instead of just the target hit by the initial impact. He also has an ability that not only can be used for some burst with some fairly good base damage, but can give him a fair abit of ability power without investing heavily into it. He then has an ultimate that can basically let him destroy a mage if he builds AD, then anyone once he stacks some AP with his Q and kills. He also has a fair bit of extra burst with a almost instant animation, if he can land the stun to use it first.
Their autoattack ranges are about the same, Ashe's maybe being only slightly longer. However when you compare the two, Veigar could be considered a superior carry to Ashe when built AD. Now when compared to Vayne late game, she would be the better carry, however that is true in almost any case situation