Before you became a member of the forum, I posted on the matter of using a food processor to make the Lehmann NY style dough at this thread: http://www.pizzamaking.com/forum/index.php/topic,2189.msg19294/topicseen.html#msg19294
. As you will see there, the biggest potential problems with food processors in making pizza dough, beyond being limited to a rather small dough batch size, are overkneading and overheating the dough. But, as noted at the above thread, both of these areas can be controlled reasonably well. I personally think a food processor is a great machine to have on hand to make pizza dough, especially for dough batch sizes that are too small for a stand mixer.
In order to increase the chewiness of a finished crust, the easiest way to do that is to move up to a higher protein flour, for example, from an all-purpose flour to a bread flour or from a bread flour to a high-gluten flour (like the KASL). You can also supplement a low-protein flour with vital wheat gluten (VWG), which will simulate, but not exactly reproduce, a higher protein/gluten flour. It is also possible to achieve increased chewiness by adding semolina to the basic flour. Remember, however, that there is a fine line between chewy and tough. So, you have to be careful how much VWG or semolina you add to the basic flour. I have written elsewhere on how much VWG to use; for the semolina, you might not want to exceed about 25% of the total flour, although I have seen formulations calling for up to 50%. My experience with high semolina levels is that the pizza slices do not respond well to reheating. The slices are leathery tough.
As you move up the protein scale, and assuming that you are using a relatively high hydration and proper kneading (i.e, not underkneading or overkneading), the gluten structure will be more substantial and better developed to retain the gases of fermentation. So, along with increased chewiness you should get a fairly good open and airy crust and crumb. As an added side benefit, you should also get a bit more crust flavor as a result of the higher protein levels. Reducing the amount of oil will increase the chewiness a bit but because the Lehmann dough formulation calls for only 1% oil, you are not likely to notice the difference. The oil will provide a bit of flavor to the finished crust and help improve the extensibility (stretchiness) of the dough, but, at 1%, these effects will be slight also.
If you think about it, the Lehmann formulation naturally has all of the above characteristics: high-gluten flour, high hydration, little oil, and no sugar. You could reduce the salt a bit to get a slight increase in the rise of the dough, but you may sacrifice some flavor in the finished crust as a result. I haven't tested this possibility before with a Lehmann dough, but may want to try it sometime just to see what happens.