  Recent experiments have shown that the phase diagrams of the kagome
superconductors $A\mathrm{V}_3\mathrm{Sb}_5$ are strongly impacted by changes
in the $c$-axis lattice parameter. Here, we show that $c$-axis deformations
impact primarily the Sb apical bonds and thus the overlap between their $p_z$
orbitals. Changes in the latter, in turn, substantially affect low-energy
electronic states with significant Sb character, most notably the central
electron pocket and the van Hove singularities located above the Fermi level.
Based on the orbital-selective character of $c$-axis strain, we argue that
these electronic states experience a non-negligible attractive electron-phonon
pairing interaction mediated by fluctuations in the apical Sb bonds. We thus
propose a multi-band model for superconductivity in
$A\mathrm{V}_3\mathrm{Sb}_5$ that includes both the Sb pocket and the V-derived
van Hove singularities. Upon comparing the theoretical phase diagram with the
experimentally observed vanishing of the $T_c$ dome across a Lifshitz
transition of the Sb pocket, we propose that either an $s^{+-}$ or an $s^{++}$
state is realized in $A\mathrm{V}_3\mathrm{Sb}_5$.
