We propose a concatenated code construction for a class of discrete-alphabet oblivious arbitrarily varying channels (AVCs) with cost constraints. The code has time and space complexity polynomial in the blocklength $n$. It uses a Reed-Solomon outer code, logarithmic blocklength random inner codes, and stochastic encoding by permuting the codeword before transmission. When the channel satisfies a condition called strong DS-nonsymmetrizability (a modified version of nonsymmetrizability originally due to Dobrushin and Stambler), we show that the code achieves a rate that for a variety of oblivious AVCs (such as classically studied error/erasure channels) match the known capacities.