I went to a Johnnie Walker Black Label promotion night about 18 months ago, and they advised that Lagavulin was indeed one of the component malts.
However, given that there are (apparently) 41 different grain & malt whiskies in JW Black, you would expect there to be more than just Caol Ila supplying the Islay content! Maybe they also use Ardbeg or Laphroiag?
It's nice to think that Diageo do their best to use only their own products in their blends, but - according to the JW legend - the recipe for JW Black has remained unchanged. (I'm very sceptical about this, since countless distilleries have closed since JW Black was first produced). In any event, back when JW Black was first conceived, it is most probable that distilleries like Lagavulin, Caol Ila, and Talisker were all owned & run by three different & independent companies.