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.