A 103 No.

There are two problems with your scenario. One is that if you throw an axe, you're not wielding it; you have no control over the weapon.
Spell Storing

A spell storing weapon allows a spellcaster to store a single targeted spell of up to 3rd level in the weapon. (The spell must have a casting time of 1 standard action.) Any time the weapon strikes a creature and the creature takes damage from it, the weapon can immediately cast the spell on that creature as a free action if the wielder desires.
The second problem is that the weapon casts the spell, not you. Thus all the temporary HP from Vampiric Touch would go to the weapon.