Yes, it can be done, it is a hassle and you will not get the same flavor as a sidewheeler. The key to smoking with a traditional stickburner with a side fire box is the draft of the cooker. It isn't just the fire source, it is also how the heat and smoke move around the meat. Every cooker has a different taste.
That being said, if you were to use a kiln to burn the wood to coals, then shoveled small amounts of coal into the Weber with the vents all open, controlling the heat with the amount of coal you add, as opposed to closing the vents, you might just get a good product. But, it will be a lot of fuss.
I'm feeling bearish, and I'm packing a Wusthof Grand Brisket slicer from MABA
"perhaps...but then again...maybe not..."