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 multiband 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 $A{\mathrm{V}}_3{\mathrm{Sb}}_5$.

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• Electron-phonon coupling
• Phase diagrams
• Quantum phase transitions
• Superconductivity
• Kagome lattice
• Strongly correlated systems