Second-moment asymptotic stabilization of a discrete-time switched stochastic system is investigated. Active operation mode of the switched system is assumed to be only periodically observed (sampled). We develop a stabilizing feedback control framework that incorporates sampled-mode-dependent time-varying feedback gains, which allow stabilization despite the uncertainty of the active operation mode between consecutive mode observation instants. We utilize the periodicity induced in the closed-loop system dynamics due to periodic mode observations, and employ discrete-time Floquet theory to obtain necessary and sufficient conditions for second-moment asymptotic stabilization of the zero solution. Furthermore, we use Lyapunov-like functions with periodic coefficients to obtain alternative stabilization conditions, which we then employ for designing feedback gains. Finally, we demonstrate the efficacy of our results with a numerical example.