Even if a starter could be made from old dough (which I'm pretty sure it can be
), it doesn't necessarily equate to making a good pizza dough or a good pie. Yeast, whether it be commerical, cake yeast, or wild yeast is only one variable amongst many require to make pizza. If everything else isn't as the shop does it, I would likely bet that your pizza dough will taste very much different from that shop despite having the same yeast culture.
To duplicate any shop's pie, you would likely have to work there or know someone who does, know the recipe and fermentation times, use the same sauce and toppings, and re-create the baking environment (or bake it in the same oven). It would be much easier just to order a pizza to go.
Even though it's fun for us hobbyist to try to reverse engineer pizzas, it is somewhat of a maddening process.
Mark for fun, just go ahead and take a tablespoon or 2 of the old dough, and mix it in a generic pizza dough of say...200gm. Let it rest at room temps covered for 12 hours. I'm willing to bet in 12 hours that dough will have risen displaying that the yeast has propagated.
Flour (100%): 118.82 g | 4.19 oz | 0.26 lbs
Water (66%): 78.42 g | 2.77 oz | 0.17 lbs
Old dough: 2 Tablespoons
Salt (1%): 1.19 g | 0.04 oz | 0 lbs | 0.25 tsp | 0.08 tbsp
Oil (1%): 1.19 g | 0.04 oz | 0 lbs | 0.26 tsp | 0.09 tbsp
Sugar (1%): 1.19 g | 0.04 oz | 0 lbs | 0.3 tsp | 0.1 tbsp
Total (170%): 202 g | 7.13 oz | 0.45 lbs | TF = N/A
Mark if you are successful in propagating this culture, I'm quite postive that using this newly acquire starter in your current recipes will yield an almost identical pie to what you are now making. It won't likely make your pizza better or worse. What it will do is maybe give you a slightly different tasting crust, but it will be very similar to what you are currently making.