This paper proposes a distributed asynchronous adaptive gradient tracking method, DASYAGT, to solve the distributed stochastic optimization problems with decision-dependent distributions over directed graphs. DASYAGT employs the local adaptive gradient to estimate the gradient of the objective function and introduces the auxiliary running-sum variable to handle asynchrony. We show that the iterates generated by DASYAGT converge, in expectation, to a stationary solution with a rate of $\mathcal{O}\left(\frac{\ln K}{\sqrt{K}}\right)$. The effectiveness of DASYAGT is further demonstrated numerically with synthetic and real-world data.