We consider a queue-channel model that captures the waiting time-dependent degradation of information bits as they wait to be transmitted. Such a scenario arises naturally in delay-sensitive communications, where information bits become useless after a certain time. Trailing the capacity results obtained recently for certain queue-channels, this paper aims to construct practical channel codes for the erasure queue-channel (EQC)---a channel characterized by highly correlated erasures, governed by the underlying queuing dynamics. Our main contributions in this paper are twofold: (i) We propose a generic `wrapper' based on interleaving across renewal blocks of the queue to convert any capacity-achieving code for a memoryless erasure channel to a capacity-achieving code for the EQC. Next, due to the complexity involved in implementing interleaved systems, (ii) we study the performance of LDPC and Polar codes without any interleaving. Motivated by the empirical performance, we show that standard Arikan's Polar transform polarizes the M/M/1 EQC. We also highlight some possible approaches and the corresponding challenges involved in proving polarization of a general EQC.