No, because "copies" of Chain of Vapor get put on the stack, you don't actually play them, it's similar to Minds desire copies not counting towards storm.
Desire might not be the best example here, since the resolution of those copies probably results in the flipped spells being cast, and indirectly resulting in a boosted storm count after all.
To use a different storm spell in the example instead:Tendrils copies do not count. Let's say you play 4 spells, and cast Tendrils. You get 5 total hits (1 original, plus storm 4). If you cast another Tendrils, this time you get 6 total hits (1 original, plus storm 5). The reason for this is exactly as Marske said: the copies go directly on the stack. Storm only counts spells that have been CAST.
