MLB offices for the most part don't use "WAR", per se. WAR is a descriptive short hand statistic and is really of little use to a front office.
But they would all have internal predictive and descriptive measurements of offensive and defensive run values for every player. Whether or not they roll everything up into one number, who knows.
And whether or not the GM and AGMs even care about internal nerd numbers, who knows. Most care. But some executives might prefer to simply classify players based on the 20-80 scale (i.e., this guy is a "role 6" player). I know of one front office that actually had their analytics department translate internal "WAR" into the 20-80 scale, for use by the executives (i.e., "the nerds think he's a role 5 player but I've seen him several times and he's just nothing more than a role 4").