I want to make a pizza on my charcoal barbecue. I saw a few videos from Weber, but none show how to get a thin crust.
What is the secret to a thin crust pizza and can it be done on a charcoal bbq?
The secret to getting a thin crust is... use less dough. It's that simple. Pizza dough is flour, water, salt, yeast: there is no magic ingredient that will make it thinner. Simply use less dough per pizza. You may find that when you first shape the dough it will try and spring back. In this case, just cover it and leave it for 5 minutes for the gluten to relax.
As for the charcoal grill, there is no reason why a thin crust pizza should be done any differently to any other pizza. It's entirely doable.
I mostly agree with Elendil's answer: most kinds of pizza dough can be made thinner simply by stretching the dough more (and pausing to rest for a minute or two if it starts to spring back). Wetter doughs will generally stretch more easily with less "spring back," but they can also rip more easily. Which you prefer is kind of up to you.
I would add three further comments:
The main "essential" aspect to any thin-crust dough is full gluten development, which can mean more kneading, using a higher gluten flour, and/or changing your fermentation process. (Longer fermentation will allow better gluten development up to a point; after that, excessively long fermentations may cause the gluten to begin to break down.) If you don't have sufficient gluten development (or have gluten breakdown due to excessive fermentation or dough acidity), your dough will rip more easily if you try to stretch it very thin.
Recipes for pizza dough vary significantly. While flour, water, salt, and yeast are the basic ingredients, dough can also include some oil, sugar, and occasionally other ingredients. There are many "traditional" opinions on what proportions are appropriate for various kinds of "thin crust" pizzas, from Neapolitan style to New York style, etc. Most pizza doughs can be stretched thin, but you should probably narrow down the style of pizza you're going for in order to determine the appropriate ingredients and proportions in your recipe.
The type of dough should also be determined by the baking process. If you can achieve a very high temperature in your grill, that will reduce your bake time, but it can also require certain dough types. For example, thin doughs baked very hot shouldn't have excess sugar, since it might cause excessive browning. There's no single answer to which dough will be "best" on a grill -- lots of people have different opinions. But you should calibrate your ingredients a bit to the temperature and time for baking.
From the phrasing of your question, it sounds like you want to put the pizza directly on the grate of your grill (without a pizza stone). If that's true, then I think the secret to using a thin crust is how you handle it as you transfer it to the grate.
To have your thin crust survive, build it on a flat board (e.g. a wooden cutting board) with enough cornmeal spread thinly under the crust so that shaking the board lets the pizza slide around. Once built, and when the barbecue is hot, hold the board over the grill, tilted down so the far side of the board almost touches the far side of the grate. Then, jiggle the board from side to side as you pull it towards you, keeping the edge of the board close to the grate. If you do it right, the pizza will slide off the board and onto the grate with very little stretching and trauma. (If you do it wrong, you'll have a cleaning job ahead of you.)