The uniqueness of Whittaker models plays an important role in the representation theory of linear reductive groups due to its relation to $L$-functions. However such uniqueness fails in general for nonlinear covering groups. We investigate this failure of uniqueness through the Gelfand-Graev representation, which is the dual of the Whittaker space. Using the pro-$p$ Iwahori-Hecke algebra, we describe the Iwahori-fixed vectors in the Gelfand-Graev representation of tame covering groups as a module over the Iwahori-Hecke algebra, generalizing work of Barbasch-Moy and Chan-Savin for linear groups. As an application we 1) compute the dimension of the space of Whittaker models for constituents of certain unramified principal series; 2) describe a connection to quantum affine Schur Weyl duality in the case of covers of $GL(r)$. This is joint work with Fan Gao and Nadya Gurevich.