We study variable-length feedback (VLF) codes under a strict delay constraint to maximize their average transmission rate (ATR) in a discrete memoryless channel (DMC) while considering periodic decoding attempts. We first derive a lower bound on the maximum achievable ATR, and confirm that the VLF code can outperform non-feedback codes with a larger delay constraint. We show that for a given decoding period, as the strict delay constraint, $L$, increases, the gap between the ATR of the VLF code and the DMC capacity scales at most on the order of $O(L^{-1})$ instead of $O(L^{-1/2})$ for non-feedback codes as shown in Polyanskiy et al. [“Channel coding rate in the finite blocklength regime,” IEEE Trans. Inf. Theory, vol. 56, no. 5, pp. 2307–2359, May 2010] . We also develop an approximation indicating that, for a given $L$, the achievable ATR increases as the decoding period decreases.