No, the artificer cannot store their specialist spells.

You've found the crux of why they can't already. No where in the specialist spells feature does it say they are added to the artificer spell list, so they aren't eligible for the Spell-Storing Item feature. Compare this to something like the extended spell list of a warlock's patrons, where the feature says:

The [patron] lets you choose from an expanded list of spells when you learn a warlock spell. The following spells are added to the warlock spell list for you. [PHB p.108, 109, & 110]

There's no subclass that uses the same wording of ERLW, but that's because for most subclasses that have extra spells (Cleric domains, Circle of Land Druids, Paladin Oaths) have a default part in the main class saying, "If you gain "a subclass" spell that doesn't appear on the [class] spell list, the spell is nonetheless a [class] spell for you."