Every part of WAR is inherently dependent on teammates and competition. Hitters don't get adjustments for opponents defense, pitchers don't get adjustments for certain elite hitters getting a day off against them, etc. However, over a 162 game schedule, discrepancies like these usually get sorted out.
As far as baserunning goes, this does have a chance to be skewed. For example, a baserunner's 1st to 3rd value may be inflated if he consistently has a hitter behind him that drives the ball to right field. The reason it's not a big deal is that baserunning values almost never exceed + 8 runs, which is less than one full WAR. The stat also passes the smell test, as Jason Heyward and Mike Trout had the highest values while Albert Pujols and Yadier Molina are at the bottom.
We know baserunning affects the value of a player, but quantifying that effect is tough. The stat used here was created by the same guy who formed UZR, and the two stats are formed similarly. That means there is some noise year-to-year, but I think it still does a good job in evaluation.