I cook pizza all the time on my BGE.
As mentioned you need lump coal not charcoal. Also get some chunks of real wood.
If you do not have any hardwood around and are a city slicker like me, Home Depot sells chunk wood. I typically have Cherry or Apple tree chunks that I use for BBQ handy so I just use that or if I have some oak or other hardwood I will use that.
The key here is you need to get things good and hot, I shoot for 500° F plus.
Also invest if a Weber Chimney for coal starting NEVER use fluid or self lighting charcoal it will ruin the flavor of your food.
I load up the chimney with coal, pretty much fill it up then put it on a turkey fryer. Look up in the top left corner of this picture and you see my chimney and turkey fryer concoction.
Next, I have a heat diffuser, I do not think this is mandatory, the grate will suit you fine.
Yes, you want to have a pizza stone, I put mine on the grill while it is warming up so it gets to temp evenly and does not crack. I never take it off I just put pizzas on and off it.
First toss the coals in and then throw the wood on top of them, leave the cover on the BGE open for a few and let everything get good and hot, I even keep the lid a little open just to encourage a good raging fire in there. I put a pair of plyers to keep the lid cracked open, once things get going good you can close it up. Then just leave all the vents 100% open. It is not hard to get 500°.
I am cooking a calzone here but you get the picture.
Pizza cook time should be about 10-15 minutes, I check under the crust
Get the huge spatula it is helpful. Here is a link to a cheap stone spatula setup. The stone is kind of small I prefer a 16" stone.
https://www.amazon.com/Cuisinart-CPS-445-Pizza-Grilling-Set/dp/B00F3BH9Y6?psc=1&SubscriptionId=AKIAILSHYYTFIVPWUY6Q&tag=duckduckgo-d-20&linkCode=xm2&camp=2025&creative=165953&creativeASIN=B00F3BH9Y6
Here is a nicer stone and below option to buy a spatula too.
Finally, the taste is awesome you get the wood fired pizza taste like you get at those brick oven pizza joints. We love it, the learning curve is not very steep it is pretty easy to do.
Good luck and enjoy!
Just found some more pics, here is the chimney at full sing, getting ready to make pizza in this instance.
One other thing, you can use parchment paper and it makes moving pizza on to grill much easier. After a few minutes you can pull it out from under the pizza and let it get crispy.
