Well...I'm going to answer my own question. Last night I experimented with my project of trying to emulate the Villaware PizzAGrill (http://www.surlatable.com/common/products/product_details.cfm?PRRFNBR=15269
). I was interested in emulating this piece of equipment for a much lower price.
1) I purchased four relatively small landscape pavers from Home Depot that have a height of 2 1/4". I propped my 14" x 16" Baking Stone on top of the pavers, which I placed on each corner. Using my Weber Genesis gas grill, I turned the three burners to their highest settings, closed the grill cover, and backed off the settings to "medium" as soon as the temperature reached 500 degrees (I was running into and out of my house so don't remember exactly how long this took--I'd say about five minutes). A comment: Although I've read numerous postings where folks try and get their ovens and grills as hot as possible, I know from experience and from reading postings about trying to make grilled pizza, that burnt crusts and undercooked toppings often result from trying to do this on a grill.
2) Because I was conducting a during the week experiment, I used store bought everything: premade corn meal dough from Trader Joe's, grated quattro stagione cheese from Trader Joe's, kalamata olives from Trader Joe's which I sliced finely, and Whole Foods 365 Brand Marinara sauce--which I smoothed out using an immersion blender and then added some finely chopped fresh garlic to.
3) Referring back to item #1, once the temperature of the grill reached 500 degrees and I backed off the heat settings to "medium," I then preheated at this approximate temperature for 30 minutes. The grill temperature ultimately increased to 550 degrees and leveled off.
4) After making my pie, I slipped it onto the stone and quickly closed the grill top.
5) After four minutes I quickly lifted the cover, inspected the pie (it was cooking nicely) and rotated it using two pairs of tongs. The left side of the Weber Genesis gas grill is definitely a hot spot, necessitating this rotation.
6) Upon closing the cover, I torqued the heat settings back to "high" until the temperature reached 550 degrees again, at which point I backed off the settings again. This only took a minute. After four minutes (or....eight minutes of total cooking time), I opened the cover and inspected the pie. It was bubbling, although not heartily. I therefore closed the cover, reset the heat settings to "high," and let it cook for one more minute. After the one minute (nine minutes of total cooking time) I turned the grill off and removed the pizza with my peel. (I will use a pair of tongs to help in this the next time as there is no "back of grill" or "back of oven" to push the peel against--I'm a neophyte at this and don't have my technique fully down yet).
The pizza was perfectly cooked. Nice crisp browned crust with the slightest (and I mean slightest) of blackened edges. I deem this experiment successful and suggest that anyone with a Weber gas grill can achieve similar if not greater levels of success. Keys: 1) get the pizza stone off the grill grate; 2) go for a nice even temperature of 500-550 degrees rather than trying to simulate a commercial oven; 3) lift the cover as little as possible but don't be too paranoid given that with proper preheating your stone will be nice and hot; 4) keep the pizza and toppings thin and minimal.