  The recent observation of high-T$_c$ superconductivity in the bilayer
nickelate La$_3$Ni$_2$O$_7$ under pressure has garnered significant interests.
While researches have predominantly focused on the role of electron-electron
interactions in the superconducting mechanism, the impact of electron-phonon
coupling (EPC) has remained elusive. In this work, we perform first-principles
calculations to study the phonon spectrum and electron-phonon coupling within
La$_3$Ni$_2$O$_7$ under pressure and explore of the interplay between EPC and
electronic interactions on the superconductivity by employing functional
renormalization group approach. Our calculations reveal that EPC alone is
insufficient to trigger superconductivity in La$_3$Ni$_2$O$_7$ under pressure.
We identify unique out-of-plane and in-plane breathing phonon modes which
selectively couple with the Ni $d_{z^2}$ and $d_{x^2-y^2}$ orbitals, showcasing
an orbital-selective EPC. Within the bilayer two-orbital model, it is revealed
that solely electronic interactions foster $s_{\pm}$-wave pairing characterized
by notable frustration in the band space, leading to a low transition
temperature. Remarkably, we find that this out-of-plane EPC can act in concert
with electronic interactions to promote the onsite and interlayer pairing in
the $d_{z^2}$ orbital, partially releasing the pairing frustration and thus
elevating T$_c$. In contrast, the inclusion of in-plane EPC only marginally
affects the superconductivity, distinct from the cuprates. Potential
experimental implications in La$_3$Ni$_2$O$_7$ are also discussed.
