Its kinda misleading, there's still animal products in this because of how the food is prepared...
Indeed, unless Burger King is using a completely separate grill for the vegetable burgers, some vegetarians/vegans are going to be upset that their vegetable patty was prepared where meat was and possibly exposed to meat, and there could be kosher/halal issues.
I believe some (or all?) of them use a Nieco ( https://nieco.com/ ) brand broiler that basically has a metal conveyor belt that carries the patty through a flame.
See this video: https://www.youtube.com/watch?v=GSuLWkBt5MI
They could, if they wanted, designate one lane solely for Impossible patties (and other non-meat products).
If they don't, looks by the time the belt makes it way back through the flame, whatever residue on it will have been mostly reduced to ash. I would guess the belt wouldn't transfer meat grease and crumbs onto a veggie patty as much as a grill would. But, after it leaves the belt, it slides down to a holding area, where it would come into contact with whatever else was on the belt before it.