  We extend the two-particle self-consistent method proposed by Vilk and
Tremblay (J. Phys. I France 7, 1309-1368 (1997)) to study superconductivity in
multi-orbital systems. Starting with the sum rules for the spin and charge
susceptibilities, we derive self-consistent equations to determine the
renormalized effective interactions. We apply this method to the two-orbital
$d_{x^2-y^2}$-$d_{3z^2-r^2}$ model for La$_2$CuO$_4$ and the five-orbital
$d$-model for LaFeAsO. Comparing the results with those of the random phase
approximation or the fluctuation exchange approximation in which vertex
corrections are ignored, we discuss how the vertex corrections affect the
pairing instability of La$_2$CuO$_4$ and the dominant pairing symmetry of
LaFeAsO.
