The invariant Einstein-Randers metrics on the quaternionic Stiefel manifold V_pHⁿ of every orthonormal p-frame in Hn are considered, V_pHⁿ is diffeomorphic to the homogeneous space Sp(n)/Sp(n-p). It's proved that there are at least two families of Sp(n)-invariant Einstein-Randers metrics on Sp(n)/Sp(n-p) for any 2 ≤ p ≤(3/4)n and eight families of invariant Einstein-Randers metrics on Sp(3)/Sp(1) and Sp(4)/Sp(2).