  In this work we investigate the exclusive production of a pair of light
neutral mesons in $e^+e^-$ annihilation, where the final state bears an even
$C$-parity. The production processes can be initiated via the photon
fragmentation or the non-fragmentation mechanism. While the fragmentation
contribution can be rigorously accounted, the non-fragmentation contributions
are calculated within the framework of collinear factorization, where only the
leading-twist light-cone distribution amplitudes (LCDAs) of mesons are
considered. Mediately solely by the non-fragmentation mechanism, the production
rates of double light neutral pseudoscalar mesons are too small to be observed
at the commissioning $e^+e^-$ facilities. In contrast, the production rates of
a pair of light neutral vector mesons are greatly amplified owing to the
significant kinematic enhancement brought by the fragmentation mechanism. It is
found that, at $\sqrt{s}=3.77$ GeV, after including the destructive
interference between the non-fragmentation and fragmentation contributions, the
production rates for $e^+e^-\to \rho^{0}\rho^{0}$ and $\rho^0\omega$ can be
lowered by about 10\% and 30\% relative to the fragmentation predictions.
Future precise measurement of these exclusive double neutral vector meson
production channels at {\tt BESIII} experiment may provide useful constraints
on the LCDAs of light vector mesons.
