You can but it will come out fairly stringy *depending* on the cut sort of like brisket. I might encourage you to use the braise or crockpot to produce bbq beef sandwiches.
I routinely sear or bbq the "london broil" cuts without issue. I think a good idea is pounding the crap out of it first and also never cook it above medium rare. By the time it rests it will be approaching the medium zone. If your serving it whole, pay attention to the grain and how you cut the slices, it makes a world of difference with the 45* angle.