  The branching fraction of $D^+\to K_{S}^{0} \pi^{0}e^+\nu_e$ is measured for
the first time using $7.93~\mathrm{fb}^{-1}$ of $e^+e^-$ annihilation data
collected at the center-of-mass energy $\sqrt{s}=3.773$~GeV with the BESIII
detector operating at the BEPCII collider, and is determined to be ${\mathcal
B}$($D^+\to K_S^0\pi^0e^+\nu_e$) = $(0.881~\pm~0.017_{\rm stat.}~\pm~0.016_{\rm
syst.})$\%. Based on an analysis of the $D^+\to K_S^0\pi^0e^+\nu_e$ decay
dynamics, we observe the $S\text{-}{\rm wave}$ and $P$-wave components with
fractions of $f_{S\text{-}{\rm wave}}$ = $(6.13~\pm~0.27_{\rm stat.}~\pm
~0.30_{\rm syst.})\%$ and $f_{\bar K^{*}(892)^0}$ = $(93.88~\pm~0.27_{\rm
stat.}~\pm~0.29_{\rm syst.})$\%, respectively. From these results, we obtain
the branching fractions ${\mathcal B}$($D^+\to (K_S^0\pi^0)_{S\text{-}{\rm
wave}}~e^+\nu_e$) = $(5.41~\pm~0.35_{\rm stat.}~\pm~0.37_{\rm
syst.})\times10^{-4}$ and ${\mathcal B}$($D^+\to \bar K^{*}(892)^0e^+\nu_e$) =
$(4.97~\pm~0.11_{\rm stat.}~\pm~0.12_{\rm syst.})$\%. In addition, the hadronic
form-factor ratios of $D^{+} \to \bar {K}^{*}(892)^0e^+\nu_e$ at $q^2=0$,
assuming a single-pole dominance parameterization, are determined to be
$r_V=\frac{V(0)}{A_1(0)}= 1.43~\pm~0.07_{\rm stat.}~\pm~0.03_{\rm syst.}$ and
$r_2=\frac{A_2(0)}{A_1(0)}=0.72~\pm~0.06_{\rm stat.}~\pm~0.02_{\rm syst.}$.
